P  n 


ilLD  H 


PLANE  TRIGONOMETRY   -. 
AND   NUMERICAL   COMPUTATION 


/~\ 


^^^^^^ 


JOHN  ALEXANDER  JAMESON,  Jr. 

1903-1934 


This  book  belonged  to  John  Alexander  Jameson,  Jr.,  A.B.,  Wil- 
liams, 1925;  B.S.,  Massachusetts  Institute  of  Technology,  1928; 
M.S.,  California,  1933.  He  was  a  member  of  Phi  Beta  Kappa,  Tau 
Beta  Pi,  the  American  Society  of  Civil  Engineers,  and  the  Sigma 
Phi  Fraternity.  His  untimely  death  cut  short  a  promising  career. 
He  was  engaged,  as  Research  Assistant  in  Mechanical  Engineering, 
upon  the  design  and  construction  of  the  U.  S.  Tidal  Model  Labora- 
tory of  the  University  of  California. 

His  genial  nature  and  unostentatious  effectiveness  were  founded 
on  integrity,  loyalty,  and  devotion.  These  qualities,  recognized  by 
everyone,  make  his  life  a  continuing  beneficence.  Memory  of  him 
will  not  fail  among  those  who  knew  him. 


PLANE  TRIGONOMETRY 

AND  NUMERICAL 

COMPUTATION 


BY 

JOHN   WESLEY   YOUNG 
»/ 

PROFESSOR    OF    MATHEMATICS 
DARTMOUTH    COLLEGE 

AND 

FRANK   MILLETT   MORGAN 

AS8I8TANT    PROFESSOR    OF    MATHEMATICS 
DARTMOUTH    COLLEGE 

r°. .ss  <+  Mi  hi  l   p  *c  is l  *  * ■  ui 

y  If 


Neta  gork  *   * 

THE   MACMILLAN   COMPANY 
1919 

All  rights  reserved 

I  C     />    « 


cat  1 


l'£&f*'  SAS35 

Copyright,  1919, 
Bv  THE  MACM1LLAN  COMPANY. 


Set  up  and  electrotyped.     Published  October,  1919. 

ENGINEERING  Uftfttjty. 


n.,1 p  (AiJiXi.i    ;..!P,...' — 

Norfoootr  Pteaa  /|    1       *  ^- 

.1.  S_  Olshinor  O.n   'Ro^T.rJnU  JL  Q^UU  <^_ 


J.  S.  Cushing  Co.  — Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

Ever  since  the  publication  of  our  Elementary  Mathematical 
Analysis  (The  Macmillan  Co.,  1917)  we  have  been  asked  by 
numerous  teachers  to  publish  separately,  as  a  textbook  in  plane 
trigonometry,  the  material  on  trigonometry  and  logarithms  of 
the  text  mentioned. 

The  present  textbook  is  the  direct  outcome  of  these  requests. 
Of  course,  such  separate  publication  of  material  taken  out  of 
the  body  of  another  book  necessitated  some  changes  and  an  in- 
troductory chapter.  As  a  matter  of  fact,  however,  we  have 
found  it  desirable  to  make  a  number  of  changes  and  additions 
not  required  by  the  necessities  of  separate  publication.  As  a 
result  fully  half  of  the  material  has  been  entirely  rewritten,  with 
the  purpose  of  bringing  the  text  abreast  of  the  most  recent 
tendencies  in  the  teaching  of  trigonometry. 

There  is  an  increasing  demand  for  a  brief  text  emphasizing  the 
numerical  aspect  of  trigonometry  and  giving  only  so  much  of  the 
theory  as  is  necessary  for  a  thorough  understanding  of  the 
numerical  applications.  The  material  has  therefore  been  ar- 
ranged in  such  a  way  that  the  first  six  chapters  give  the  essen- 
tials of  a  course  in  numerical  trigonometry  and  logarithmic 
computation.  The  remainder  of  the  theory  usually  given  in 
the  longer  courses  is  contained  in  the  last  two  chapters. 

More  emphasis  than  hitherto  has  been  placed  on  the  use  of 
tables.  For  this  purpose  a  table  of  squares  and  square  roots 
has  been  added.  Recent  experience  has  emphasized  the  appli- 
cations of  trigonometry  in  navigation.  We  have  accordingly 
added  some  material  in  the  text  on  navigation,  have  introduced 

v 

889757 


vi  PREFACE 

the  haversine,  and  have  added  a  four-place  table  of  haversines 
for  the  benefit  of  those  teachers  who  feel  that  the  use  of  the 
haversine  in  the  solution  of  triangles  is  desirable.  This  material 
can,  however,  be  readily  omitted  by  any  teacher  who  prefers 
to  do  so. 

J.  W.  Young, 
F.  M.  Morgan. 

Hanover,  N.H., 
August,  1919. 


CONTENTS 

CHAPTER  PAGES 

I.     Introductory  Conceptions 1-10 

II.     The  Right  Triangle       .        .        .        .        .        .  11-31 

III.  Simple  Trigonometric  Relations  .         .        .         .  32-39 

IV.  Oblique  Triangles .  40-49 

V.     Logarithms 50-60 

VI.     Logarithmic  Computation 61-74 

VII.     Trigonometric  Relations 75-87 

VIII.     Trigonometric  Relations  (continued)    .         .         .  88-103 

Tables 106-119 

Index 121-122 


vu 


PLANE   TRIGONOMETRY   AND 
NUMERICAL   COMPUTATION 

CHAPTER  I 
INTRODUCTORY  CONCEPTIONS 

1.  The  Uses  of  Trigonometry.  The  word  "  trigonometry  " 
is  derived  from  two  Greek  words  meaning  "  the  measurement 
of  triangles."  A  triangle  has  six  so-called  elements  (or  parts)  • 
viz.,  its  three  sides  and  its  three  angles.'  AV'e v*rksw  from  our 
study  of  geometry  that,  in  general,  if  three  elements  of  a  tri- 
angle (not  all  angles)  are  given,  the  triangle  is  completely 
determined.*  Hence,  if  three  such  determining  elements  of  a 
triangle  are  given,  it  should  be  possible  to  compute  the  remain- 
ing elements.  The  methods  by  which  this  can  be  done,  i.e. 
methods  for  "  solving  a  triangle,"  constitute  one  of  the  prin- 
cipal objects  of  the  study  of-  trigonometry. 

If  two  of  the  angles  of  a  triangle  are  given,  the  third  angle 
can  be  found  from  the  relation  A  +  B  -f-  C  =  180°  (A,  B,  and 
C  representing  the  angles  of  the  triangle)  ;  also,  in  a  right  tri- 
angle, if  two  of  the  sides  are  known,  the  third  side  can  be 
found  from  the  relation  a2  +  b2  =  c-  (a,  b  being  the  legs  and  c 
the  hypotenuse).  But  this  is  nearly  the  limit  to  which  the 
methods  of  elementary  geometry  will  allow  us  to  go  in  the 
solution  of  a  triangle. 

Trigonometry  f   is   the   foundation  of  the  art  of  surveying 

*  What  exceptions  are  there  to  this  statement  ? 

t  Throughout  this  book  we  shall  confine  ourselves  to  the  subject  of  "plane 
trigonometry,"  which  deals  with  rectilinear  triangles  in  a  plane.  "  Spherical 
trigonometry"  deals  with  similar  problems  regarding  triangles  on  a  sphere 
whose  sides  are  arcs  of  great  circles. 

B  1 


PLANE  TRIGONOMETRY 


H, 


and  of  much  of  the  art  of  navigation.  It  is,  moreover,  of 
primary  importance  in  practically  every  branch  of  pure  and 
applied  mathematics.  Many  of  the  more  elementary  applica- 
tions will  be  presented  in  later  portions  of  this  text. 

2.  The  "  Shadow  Method."  The  ancient  Greeks  employed 
the  theory  of  similar  triangles  in  the  solution  of  a  special  type 
of  triangle  problem  which  it  is  worth  our  while  to  examine 
briefly,  because  it  contains  the  germ  of  the  theory  of  trigo- 
nometry. 

It  is  desired  to  find  the  height  CA  of  a  vertical  tower  stand- 
ing on  a  level  plain.  It  is  observed 
that  at  a  certain  time  the  tower  casts  a 
shadow  42  ft.  long.  At  the  same  time 
a  pole  C'A',  10  ft.  long,  held  vertically 
with  one  end  on  the  ground  casts  a 
shadow  7  ft.  long.  From  these  data 
the  height  of  the  tower  is  readily  com- 
puted as  follows :  The  right  triangles 
ABC  and  A'B'C  are  similar  since  Z  B 
=  Z  B'.     (Why  ?)     Therefore  we  have 

CA=C'A'     10 
BC 


A 

A: 

B'  7    C' 


or 


CA  = 


B'C 
C'A' 


B'C 
The  tower  is  then  60  ft.  high. 

3.  A  "  Function  "  of  an  Angle. 


£<7  =  y  x42  =  60. 


From  the  point  of  view  of 
our  future  study  the  important  thing  to  notice  in  the  solution 

CA    C'A' 

of  the  preceding  article  is  the  fact  that  the  ratios  ,  — — 

v  Bkj    b  g 

are  equal,  i.e.  that  the  ratio  of  the  side  opposite  the  angle  B  to 

the  side  adjacent  to  the  angle  is  determined  by  the  size  of  the  angle, 

and  does  not  depend  at  all  on  any  of  the  other  elements  of  the 

triangle,  provided  only  it  is  a  right  triangle. 


I,  §  3]  INTRODUCTORY  CONCEPTIONS  3 

Definition.  Whenever  a  quantity  depends  for  its  value  on 
a  second  quantity,  the  first  is  called  a  function  of  the  second. 

Thus  in  our  example  the  ratio  of  the  side  opposite  an  angle 
of  a  right  triangle  to  the  side  adjacent  is  a  quantity  which 
depends  for  its  value  only  on  the  angle ;  it  is,  therefore,  called 
a  function  of  the  angle.  This  ratio  is  merely  one  of  several 
functions  of  an  angle  which  we  shall  define  in  the  next 
chapter.  By  means  of  these  functions  the  fundamental  prob- 
lem of  trigonometry  can  be  readily  solved. 

The  particular  function  which  we  have  discussed  is  called 
the  tangent  of  the  angle.  Explicitly  defined  for  an  acute  angle 
of  a  right  triangle,  we  have 

tangent  of  angle  =  ^ide  opposite  the  angle_. 
side  adjacent  to  the  angle 

If  the  angle  B  in  the  preceding  example  were  measured  it 
would  be  found  to  contain  55°.  In  any  right  triangle  then 
containing  an  angle  of  55°  we  should  find  this  ratio  to  be  equal 
to  -T0-,  or  1.43.  If  the  angle  is  changed,  this  ratio  is  changed, 
but  it  is  fixed  for  any  given  angle.  If  the  angle  is  45°,  the 
tangent  is  equal  to  1,  since  in  that  case  the  triangle  is 
isosceles. 

The  word  tangent  is  abbreviated  "  tan."  Thus  we  have 
already  found  tan  55°  =  1.43  and  tan  45°  =  1.00.  Similarly 
to  every  other  acute  angle  corresponds  a  definite  number, 
which  is  the  tangent  of  that  angle.  The  values  of  the  tan- 
gents of  angles  have  been  tabulated.  ^Ve  shall  have  occasion 
to  use  such  tables  extensively  in  the  future.        \ 

If  a,  6,  c  are  the  sides  of  a  right  triangle  ABC  with  right 
angle  at  C  and  with  the  usual  notation  whereby  the  side  a  is 
opposite  the  angle  A  and  side  b  opposite  the  angle  B,  the  defi- 
nition of  the  tangent  gives 

tanjB  =  -. 
a 


PLANE  TRIGONOMETRY 


[I,  §3 


From  this  we  get  at  once, 


b  =  a  tan  B    and     a  = 


tan  B 

These  are  our  first  trigonometric  formulas.  By  means  of 
them  and  a  table  of  tangents  we  can  compute  either  leg  of  a 
right  triangle,  if  the  other  leg  and  an  acute  angle  are  given. 


EXERCISES 

1.  What  is  meant  by  "the  elements  of  a  triangle  "  ?  by  "  solving  a 
triangle ' '  ? 

2.  A  tree  casts  a  shadow  20  ft.  long,  when  a  vertical  yardstick  with 
one  end  on  the  ground  casts  a  shadow  of  2  ft.     How  high  is  the  tree  ? 

3.  A  chimney  is  known  to  be  90  ft.  high.  How  long  is  its  shadow 
when  a  9-foot  pole  held  vertically  with  one  end  on  the  ground  casts  a 
shadow  5  ft.  long  ? 

4.  Give  examples  from  your  own  experience  of  quantities  which  are 
functions  of  other  quantities. 

5.  Define  the  tangent  of  an  acute  angle  of  a  right  triangle.  Why  does 
its  value  depend  only  on  the  size  of  the  angle  ? 

6.  In  the  adjacent  figure  think  of  the  line  BA  as  rotating  about  the 
point  B  in  the  direction  of  the  arrow,  starting  from 
the  position  BC  (when  the  angle  B  is  0)  and  assum- 
ing successively  the  positions  BAh  BA%,  BA3, 

Show  that  the  tangent  of  the  angle  B  is  very 
small  when  B  is  very  small,  that  tan  B  increases  as 
the  angle  increases,  that  tan  B  is  less  than  1  as 
long  as  B  is  less  than  45°,  that  tan  45°  =  1,  that 
tan  B  is  greater  than  1  if  the  angle  is  greater  than 
45°,  and  that  tan  B  increases  without  limit  as  B  ap- 
proaches 90°. 

7.   The  following  table  gives  the  values  of  the  tan- 
gent for  certain  values  of  the  angle  i 


angle 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

tangent 

0.176 

0.364 

0.577 

0.839 

1.19 

1.73 

2.75 

I,  §4] 


INTRODUCTORY  CONCEPTIONS 


// 


(9) 


60° 
20° 


By  means  of  this  table  find  the  other  leg  of  a  right  triangle  ABC  from 
the  elements  given : 

^  (a)  B  =  50°,  a  =  10  (d)  B  =  20°,  b  =  13 
(6)  B  =  70°,  a  =  16  (e)  A  =  30°,  6=5 
(c)   B  =  40°,  &  =  24        (/)  A  =  10°,  &  =  62 

8.  From  the  data  and  the  results  of  the  preceding  exercise  find  the 
other  acute  angle  and  the  hypotenuse  of  each  of  the  right  triangles. 

^4.  Coordinates  in  a  Plane.  The  student  should  already  be 
familiar  from  his  study  of  algebra  with  the  method  of  locating 
points  in  a  plane  by  means  of  coordinates.  Since  we  shall 
often  have  occasion  to  use  such  a  method  in  the  future,  we  will 
recall  it  briefly  at  this  point. 

The  method  consists  in  referring  the  points  in  question  to 
two  straight  lines  X'X  and  Yl  Y,  at  right  angles  to  each  other, 
which  are  called  the  axes  of 

Coordinates.       X'X    is    USUally      Second  Quadrant 

drawn  horizontally  and  is 
called  the  x-axis  ;  Y'  Y,  which 
is  then  vertical,  is  called  the 
y-axis. 

The  position  of  any  point 
P  is  completely  determined 
if  its  distance  (measured  in 
terms  of  some  convenient 
unit)  and  its  direction  from  each  of  the  axes  is  known.  Thus 
the  position  of  Px  (Fig.  2)  is  known,  if  we  know  that  it  is  4 
units  to  the  right  of  the  ?/-axis  and  2  units  above  the  x-axis.  If 
we  agree  to  consider  distance  measured  to  the  right  or  upwards 
as  positive,  and  therefore  distance  measured  to  the  left  or  down- 
ward as  negative ;  and  if,  furthermore,  we  represent  distances 
and  directions  measured  parallel  to  the  x-axis  by  x,  and  distances 
and  directions  measured  parallel  to  the  y-axis  by  y,  then  the 
position  of  Px  may  be  completely  given  by  the  specifications 
»  =  -r-4,  2/=-|-2;  or  more  briefly  still  by  the  symbol  (4,  2). 


M, 


X' 


Ms    O 


Third  Quadrant 


Mt 


Mt 


rl  Fourth  Quadrant 


Fig.  2 


6  PLANE  TRIGONOMETRY  [I,  §  4 

Similarly,  the  point  P2  in  Fig.  2  is  completely  determined 
by  the  symbol  (-3,  5).  Observe  that  in  such  a  symbol  the  x  of 
the  point  is  written  first,  the  y  second.  The  two  numbers  x 
and  y,  determining  the  position  of  a  point,  are  called  the 
coordinates  of  the  point,  the  x  being  called  the  x-coordinate 
or  abscissa,  the  y  being  called  the  y-coordinate  or  ordinate 
of  the  point.  What  are  the  coordinates  of  P3  and  PA  in 
Fig.  2? 

The  two  axes  of  coordinates  divide  the  plane  into  four  regions 
called  quadrants,  numbered  as  in  Fig.  2.  The  quadrant  in 
which  a  point  lies  is  completely  determined  by  the  signs  of  its 
coordinates.  Thus  points  in  the  first  quadrant  are  character- 
ized by  coordinates  (+,  -+-),  those  in  the  second  by  (  — ,  +), 
those  in  the  third  by  (  — ,  —  ),  and  those  in  the  fourth  by  (-f,  — ). 

Square-ruled  paper  (so-called  coordinate  or  cross  section 
paper)  is  used  to  advantage  in  "  plotting  "  (i.e.  locating)  points 
by  means  of  their  coordinates. 

5.  Magnitude  and  Directed  Quantities.  In  the  last  article 
we  introduced  the  use  of  positive  and  negative  numbers,  i.e. 
the  so-called  signed  numbers,  while  in  the  preceding  articles, 
where  we  were  concerned  with  the  sides  and  angles  of  triangles, 
we  dealt  only  with  unsigned  numbers.  The  latter  represent 
magnitude  or  size  only  (as  a  length  of  20  ft.),  while  the  former 
represent  both  a  magnitude  and  one  of  two  opposite  direc- 
tions or  senses  (as  a  distance  of  20  ft.  to  the  left  of  a  given 
line).  We  are  thus  led  to  consider  two  kinds  of  quantities  : 
(1)  magnitudes,  and  (2)  directed  quantities.  Examples  of  the 
former  are :  the  length  of  the  side  of  a  triangle,  the  weight  of 
a  barrel  of  flour,  the  duration  of  a  period  of  time,  etc.  Ex- 
amples of  the  latter  are :  the  coordinates  of  a  point,  the  tem- 
perature (a  certain  number  of  degrees  above  or  below  zero), 
the  time  at  which  a  certain  event  occurred  (a  certain  number 
of  hours  before  or  after  a  given  instant),  etc. 


I,  §  6]  INTRODUCTORY  CONCEPTIONS  7 

Geometrically,  the  distinction  between  directed  quantities 
and  mere  magnitudes  corresponds  to  the  fact  that,  on  the  one 
hand,  we  may  think  of  the  line  segment  AB  as  drawn  from  A  to 
B  or  from  B  to  A ;  and,  on  the  other  hand,  we 
may  choose  to  consider  only   the  length  of      '    '    *~~*    '     ' 
such  a  segment,  irrespective  of  its  direction. 
Figure  3  exhibits  the  geometric  representation 
of  5,  +  5,  and  —  5.     A  segment  whose  direc- 
tion is  definitely  taken  account  of  is  called  ^'directed  segment. 
The  magnitude  of  a  directed  quantity  is  called  its  absolute 
value.     Thus  the  absolute  value  of  —  5  (and  also  of  +  5)  is  5. 
Observe  that  the  segments  OMu  MXPX  (Fig.  2)  representing 
the  coordinates  of  Px  are  directed  segments. 

6.  Directed  and  General  Angles.  In  elementary  geometry 
an  angle  is  usually  defined  as  the  figure  formed  by  two  half- 
lines  issuing  from  a  point.  However,  it  is  often  more  serviceable 
to  think  of  an  angle  as  being  generated 
by  the  rotation  in  a  plane  of  a  half-line 
OP  about  the  point  0  as  a  pivot,  start- 
ing from  the  initial  position  OA  and 
ending  at  the  terminal  position  OB  (Fig. 
4).  We  then  say  that  the  line  OP  has 
generated  the  angle  AOB.  Similarly,  if  OP  rotates  from  the 
initial  position  OB,  to  the  terminal  position  OA,  then  the  angle 
BOA  is  said  to  be  generated.  Considerations  similar  to  those 
regarding  directed  line  segments  (§  5)  lead  us  to  regard  one  of 
the  above  directions  of  rotation  as  positive  andx  the  other  as 
negative.  It  is  of  course  quite  immaterial  which  one  of  the 
two  rotations  we  regard  as  positive,  but 
we  shall  assume,  from  now  on,  that 
counterclockwise  rotation  is  positive  and 
clockwise  rotation  is  negative. 

Still  another  extension  of  the  notion  Fig.  5 


8 


PLANE  TRIGONOMETRY 


[I,  §6 


of  angle  is  desirable.  In  elementary  geometry  no  angle  greater 
than  360°  is  considered  and  seldom  one  greater  than  180°.  But 
from  the  definition  of  an  angle  just  given,  we  see  that  the 
revolving  line  OP  may  make  any  number  of  complete  revolu- 
tions before  coming  to  rest,  and  thus  the  angle  generated  may 
be  of  any  magnitude.  Angles  generated  in  this  way  abound 
in  practice  and  are  known  as  angles  of  rotation  * 

When  the  rotation  generating  an  angle  is  to  be  indicated,  it  is 
customary  to  mark  the  angle  by  means  of  an  arrow  starting  at 
the  initial  line  and  ending  at  the  terminal  line.  Unless  some 
such  device  is  used,  confusion  is  liable  to  result.     In  Fig.  G 


30° 


390' 


750 


1110 


Fig.  (5 


angles  of  30°,  390°,  750°,  1110°,  are  drawn.     If  the  angles  were 
not  marked  one  might  take  them  all  to  be  angles  of  30°. 

7.  Measurement  of  Angles.  For  the  present,  angles  will  be 
measured  as  in  geometry,  the  degree  (°)  being  the  unit  of  measure.  A 
complete  revolution  is  360°.  The  other  units  in  this  system  are  the 
minute  ('),  of  which  60  make  a  degree,  and  the  second  ("),  of  which  60 
make  a  minute.  This  system  of  units  is  of  great  antiquity,  having  been 
used  by  the  Babylonians.  The  considerations  of  the  previous  article  then 
make  it  clear  that  any  real  number,  positive  or  negative,  may  represent  an 
angle,  the  absolute  value  of  the  number  representing  the  magnitude  of 
the  angle,  the  sign  representing  the  direction  of  rotation. 


v 


Fig.  7 


Consider  the  angle  XOP  =  0,  whose  vertex  O  coincides  with  the  origin 
0  of  a  system  of  rectangular  coordinates,  and  whose  initial  line  OX  coin- 

*For  example,  the  minute  hand  of  a  clock  describes  an  angle  of  —180° 
n  30  minutes,  an  angle  of  —  540°  in  90  minutes,  and  an  angle  of  —  720°  in  120 
ninutes. 


I,  §  8]  INTRODUCTORY  CONCEPTIONS  9 

cides  with  the  positive  half  of  the  a;-axis  (Fig.  7) .  The  angle  0  is  then 
said  to  be  in  the  first,  second,  third,  or  fourth  quadrant,  according  as  its 
terminal  .line  OP  is  in  the  first,  second,  third,  or  fourth  quadrant. 

8.  Addition  and  Subtraction  of   Directed   Angles.    The 

meaning  to  be  attached  to  the  sum  of  two  directed  angles  is  analogous  to 
that  for  the  sum  of  two  directed 

line  segments.     Let  a  and  b  be  /*  /& 

two  half-lines  issuing  from  the  /  £, 


Y 


same  point  O  and  let  (ab)  repre- 
sent an  angle  obtained  by  rotat-      j£FSJ — ' 5~  q 

ing  a  half -line  from  the  position  jrIG#  y 

a  to  the  position  b.     Then  if  we 

have  two  angles  (a&)  and  (6c)  with  the  same  vertex  O,  the  sum  (a6)  +  (6c) 
of  the  angles  is  the  angle  represented  by  the  rotation  of  a  half -line  from 
the  position  a  to  the  position  b  and  then  rotating  from  the  position  b  to  the 
position  c.  But  these  two  rotations  are  together  equivalent  to  a  single  rota- 
tion from  a  to  c,  no  matter  what  the  relative  positions  of  a,  6,  c  may  have 
been.  Hence,  we  have  for  any  three  half -lines  a,  b,  c  issuing  from  a  point  0, 
(1)  (ab)  +  (bc)=(ac),  (ob)  +  (bc)=0,  (ab)  =  (cb)-(ca). 
It  must  be  noted,  however,  that  the  equality  sign  here  means  "  equal, 
except  possibly  for  multiples  of  360V  The  proof  of  the  last  relation  is  > 
left  as  an  exercise.  ^^ 

EXERCISES  \^\) 

1.  On  square-ruled  paper  draw  two  axes  of  reference  and  then  plot  the 
following  points:  (2,  3),  (-  4,  2),  (-  7,  -  1),  (0,  -  3),  (2,  -  5),  (5,  0). 

2.  What  are  the  coordinates  of  the  origin  ? 

3.  Where  are  all  the  points  for  which  x  —  2?  x  =—  3  ?  y  —  —  1  ? 
y  =  ±?  x  =  0? 

4.  Show  that  any  point  P  on  the  2/-axis  has  coordinates  of  the  form 
(0,  y) .     What  is  the  form  of  the  coordinates  of  any  point  on  the  x-axis  ? 

5.  A  right  triangle  has  the  vertex  of  one  acute  angle  at  the  origin  and 
one  leg  along  the  se-axis.  The  vertex  of  the  other  acute  angle  is  at 
(7,  10).     What  is  the  tangent  of  the  angle  at  O  ?    *?  -\ 

6.  What  angle  does  the  minute  hand  of  a  clock  describe  in  2  hours 
and  30  minutes  ?  in  4  hours  and  20  minutes  ?    '  \  /  a 

7.  Suppose  that  the  dial  of  a  clock  is  transparent  so  that  it  may  be 
read  from  both  sides.  Two  persons  stationed  at  opposite  sides  of  the  dial 
observe  the  motion  of  the  minute  hand.  In  what  respect  will  the  angles 
described  by  the  minute  hand  as  seen  by  the  two  persons  differ? 


10  PLANE  TRIGONOMETRY  [I,  §  8 

i         X  / 

4  8.    In  what  quadrants  are  the  following  angles  :  87°  ?  135°  ?  —  325°  ? 

540°?  1500°?  -270°? 

9.  In  what  quadrant  is  0/2  if  0  is  a  positive  angle  less  than  360°  and  in 
the  second  quadrant  ?  third  quadrant  ?  fourth  quadrant  ? 

10.  By  means  of  a  protractor  construct  27°  +  85°  +  (—  30°)  +  20°  + 
(-45°). 

11.  By  means  of  a  protractor  construct  —  130°  +  56°  —  24°. 


I 


J 


CHAPTER   II 

THE  RIGHT  TRIANGLE 

9.  Introduction.  At  the  beginning  of  the  preceding  chap- 
ter we  described  the  fundamental  problem  of  trigonometry  to 
be  the  "  solution  of  the  triangle,"  i.e.  the  problem  of  com- 
puting the  unknown  elements  of  a  triangle  when  three  of  the 
elements  (not  all  angles)  are  given.  This  problem  can  be 
solved  by  finding  relations  between  the  sides  and  angles  of  a 
triangle  by  means  of  which  it  is  possible  to  express  the  un- 
known elements  in  terms  of  the  known  elements.  In  order 
to  establish  such  relations,  it  has  been  found  desirable  to 
define  certain  functions  of  an  angle.  One  such  function  —  the 
tangent  —  was  introduced  in  §  3  by  way  of  preliminary  illus- 
tration. 

In  the  present  chapter,  we  shall  give  a  new  definition  of  the 
tangent  of  an  angle  and  also  define  two  other  equally  impor- 
tant functions  —  the  sine  and  the  cosine.  It  should  be  noted 
that  the  definition  given  for  the  tangent  in  §  3  applies  only  to 
an  acute  angle  of  a  right  triangle.  For  the  purposes  of  a  sys- 
tematic study  of  trigonometry  we  require  a  more  general  defini- 
tion, which  will  apply  to  any  angle,  positive  or  negative,  and 
of  any  magnitude.  Such  definitions  are  given  in  the  next 
article,  in  which  the  notion  of  a  system  of  coordinates  plays  a 
fundamental  role,  the  notion  of  a  triangle  not  being  introduced 
at  all.  After  considering  some  of  the  consequences  of  our 
definitions  in  §§  11-13,  we  consider  the  way  in  which  these 
definitions  enable  us  to  express  relations  between  the  sides 
and  angles  of  a  right  triangle.  These  results  are  then  imme- 
diately applied  to  the  solution  of  numerical  problems  by  means 
r  of  tables  and  to  applications  in  surveying  and  navigation. 

11 


12 


PLANE  TRIGONOMETRY 


[II,  §  10 


10.   The   Sine,   Cosine,   and  Tangent  of  an  Angle.     We 

may  now  define  three  of  the  functions  referred  to  in  §  3.     To 
this  end  let  0  =  XOP  (Fig.  9)  be  any  directed  angle,  and  let 


zyL 


us  establish  a  system  of  rectangular  coordinates  in  the  plane 
of  the  angle  such  that  the  initial  side  OX  of  the  angle  is  the 
positive  half  of  the  sc-axis,  the  vertex  0  being  at  the  origin  and 
the  y-axis  being  in  the  usual  position  with  respect  to  the 
#-axis.  Let  the  units  on  the  two  axes  be  equal.  Finally,  let 
P  be  any  point  other  than  0  on  the  terminal  side  of  the  angle 
6,  and  let  its  coordinates  be  (x,  y).  The  directed  segment 
OP  =  r  is  called  the  distance  of  P  and  is  always  chosen  posi- 
tive. The  coordinates  x  and  y  are  positive  or  negative  accord- 
ing to  the  conventions  previously  adopted.     We  then  define 


The  sine  of  8  = 
The  cosine  of  6  = 


ordinate  of  P  _  y 
distance  of  P  ~  r 
abscissa  of  P     x 


distance  of  P 


™*     ,  *  /v      ordinate  of  P     y  . .    _ 

The  tangent  of  8  =  -r — -. j-p=~,  provided  x  =£  0.* 

These  functions  are  usually  written  in  the  abbreviated  forms 
sin  0,  cos  0,  tan  0,  respectively ;  but  they  are  read  as  "  sine  0" 
"  cosine  0,"  "  tangent  0."  It  is  very  important  to  notice  that 
the  values  of  these  functions  are  independent  of  the  position 
of  the  point  P  on  the  terminal  line.  For  let  P'  (x\  y')  be  any 
other  point  on  this  line.  Then  from  the  similar  right  triangles 
xyrf  and  x'y'r1  it  follows  that  the  ratio  of  any  two  sides 
of  the  triangle  xyr  is  equal  in   magnitude  and   sign   to   the 

*  Prove  that  x  and  y  cannot  be  zero  simultaneously. 

t  Triangle  xyz  means  the  triangle  whose  sides  are  x,  y,  z. 


II,  §  11] 


THE  RIGHT  TRIANGLE 


13 


ratio  of  the  corresponding  sides  of  the  triangle  x'y'r'.  There- 
fore the  values  of  the  functions  just  defined  depend  merely 
on  the  angle  9.  They  are  one-valued  functions  of  6  and  are 
called  trigonometric  functions. 

Since  the  values  of  these  functions  are  defined  as  the  ratios 
of  two  directed  segments,  they  are  abstract  numbers.  They 
may  be  either  positive,  negative,  or  zero.  Remembering  that  r 
is  always  positive,  we  may  readily  verify  that  the  signs  of  the 
three  functions  are  given  by  the  following  table. 


Quadrant 

Sine 

Cosine 

Tangent 

1 
•  + 

+ 

2 

3 

+ 

4 

+ 

11.  Values  of  the  Functions  for  45°,  135°,  225°,  315°.    In 

each  of  these  cases  the  triangle  xyr  is  isosceles.  Why? 
Since  the  trigonometric  functions  are  independent  of  the 
position  of  the  point  P  on  the  terminal  line,  we  may  choose 
the  legs  of  the  right  triangle  xyr  to  be  of  length  unity,  which 


M.    &i 


C^-L 


'^%\- 


Fig.  10 


gives  the  distance  OP  as  V2.     Figure  10  shows  the  four  angles 
with  all  lengths  and  directions  marked.     Therefore, 

1 


sin   45°=  ---, 

V2 

cos    45°  = 

sin  135°  =  —  , 
V2 

cos  135°  = 

sin  225°  = — , 

V2 

cos  225°  = 

sin  315°  =  -—, 

cos  315°  = 

V2 


1 

i 
i 

V2 


tan   45°  =  1, 
tan  135°  =  -1, 
tan  225°  =  1, 
tan  315°  =  -  1. 


14 


PLANE  TRIGONOMETRY 


[II,  §  12 


12.  Values  of  the  Functions  for  30°,  150°,  210°,  330°.  From 
geometry  we  know  that  if  one  angle  of  a  right  triangle  con- 
tains 30°,  then  the  hypotenuse  is  double  the  shorter  leg, 
which  is  opposite  the  30°  angle.  Hence  if  we  choose  the 
shorter   leg  (ordinate)   as   1,  the  hypotenuse  (distance)  is  2, 


Ml    'I<s^L 


vz 


•vT 


dLL± 


t» 


Fig.  11 


and  the  other  leg  (abscissa)  is  V3.  Figure  11  shows  angles  of 
30°,  150°,  210°,  330°  with  all  lengths  and  directions  marked. 
Hence  we  have 

cos    30°=-^,  tan    30*  =  — , 

2  '  V3 


sin    W-;|, 


sin  150°  =  ^, 


sin  210°  = 


2' 


sin  330°  =  -  -, 

2' 


cos  150°  =  -  ^?,  tan  150°  =  -  — , 

2  V3 


cos  210°  = 


V3 

2   ' 


cos  330c 


V3 
2  : 


tan  210°  = 


V3J 


tan  330°  =  - 


V3 


13.   Values  of  the  Functions  for  60°,  120°,  240°,  300°.     It  is 

left  as  an  exercise  to  construct  these  angles  and  to  prove  that 


sin    60°  =  ^5, 


cos    60c 


sin  120°  =  -^, 

cos  120°  =  --, 

2' 

sin  240°=-^?, 

.     2   ' 

cos  240°  =  --, 

2' 

sin  300°  =  -^, 

2 

cos  300°  =1, 

tan    60°=V3, 
tanl20°  =  -V3, 
tan240°=V3, 
tan  300°  =  -  V3. 


II,  §  14] 


THE  RIGHT  TRIANGLE 


15 


14.  Sides  and  Angles  of  a  Right  Triacgle.  Evidently  any 
right  triangle  ABC  can  be  so  placed  in  a  system  of  coordi- 
nates that  the  vertex  of  either  acute 
angle  coincides  with  the  origin  O 
and  that  the  ad'jacent  leg  lies  along 
the  positive  end  OX  of  the  aj-axis 
(Fig.  12).  The  following  relations 
then  follow  at  once  from  the  defini- 
tions of  the  sine,  cosine,  and  tangent 
of  §  10. 

In  any  right  triangle,  the  trigonometric  functions  of  either  acute 
angle  are  given  by  the  ratios : 


the  sine 


the  cosine  = 


side  opposite  the  angle 

hypotenuse 
side  adjacent  to  the  angle 


the  tangent 


hypotenuse 
side  opposite  the  angle 


side  adjacent  to  the  angle ' 
These  relations  are  fundamental  in  all  that  follows.  They 
should  be  firmly  fixed  in  mind  in  such  a  way  that  they  can  be 
readily  applied  to  any  right  triangle  in  what- 
ever position  it  may  happen  to  be  (for  example 
as  in  Fig.  13).  The  student  should  be  able  to 
reproduce  any  of  the  following  relations  with- 
out hesitation  whenever  called  for.  They 
should  not  be  memorized,  but  should  be  read 
from  an  actual  or  imagined  figure : 

b 


Fig.  13 


sin^l 


cos  A 


sin  B 


cos  B=-, 
c 


tan  A  =  - ,   tan  B  = 


Also  the  known  relation : 


C2  =  a2  +  b2. 


16 


PLANE  TRIGONOMETRY 


[II,  §  14 


If  any  two  elements  (other  than  the  right  angle)  of  a  right 
triangle  are  given,  we  can  then  find  a  relation  connecting  these 
two  elements  with  any  unknown  element,  from  which  relation 
the  unknown  element  can  be  computed. 


15.  Applications.  The  angle  which  a  line  from  the  eye  to 
an  object  makes  with  a  horizontal  line  in  the  same  vertical 
plane  is  called  an  angle  of  elevation  or  an  angle  of  depression, 


Horizontal 


Fig.  14 


according  as  the  object  is  above  or  below  the  eye  of.  the  ob- 
server (Fig.  14).     Such  angles  occur  in  many  examples. 

Example  1.  A  man  wishing  to  know  the  distance  between  two  points 
A  and  B  on  opposite  sides  of  a  pond  locates  a  point  C  on  the  land  (Fig. 
15)  such  that  AC  =  200  rd.,  angle  C  =  30°,  and  angle  B  =  90°.  Find  the 
distance  AB. 

AB 
AG 

AB  =  AC  sin  G 
=  200  •  sin  30° 

100  rd. 


Solution  : 


sin  C.         (Why  ?) 


=  200  •  * 


Fig.  15 


Example  2.  Two  men  stationed  at  points  A  and  G  800  yd.  apart  and 
in  the  same  vertical  plane  with  a  balloon  B,  observe  simultaneously  the 
angles  of  elevation  of  the  balloon  to  be  30°  and  45°  respectively.  Find  the 
height  of  the  balloon. 

Solution  :  Denote  the  height  of  the  balloon  DB  by  y,  and  let  DC  =  x; 
then  AD  =  800  -  x. 

L 


800-x  D       x 

Fig.  16 


II,  §  15J  THE  RIGHT  TRIANGLE  17 

Since  tan  45°  =  1,  we  have  1  =-, 

x 

1  y 

and  since  tan  30°  =s  1/V3,  we  have  — - ==  —  g^  _  x' 

Therefore  x  =  y  and  800  —  x  —  y  V3. 

800 
Solving  these  equations  for  y,  we  have  y  — =  292.8  yd. 

V3  +  1 

EXERCISES 

•  1.  In  what  quadrants  is  the  sine  positive  ?  cosine  negative  ?  tangent 
positive  ?  cosine  positive?  tangent  negative  ?  sine  negative  ? 

2.  In  what  quadrant  does  an  angle  lie  if 

(a)  its  sine  is  positive  and  its  cosine  is  negative  ? 

(6)  its  tangent  is  negative  and  its  cosine  is  positive? 

(c)  its  sine  is  negative  and  its  cosine  is  positive  ? 

(d)  its  cosine  is  positive  and  its  tangent  is  positive  ? 

3.  Which  of  the  following  is  the  greater  and  why  :  sin  49°  or  cos  49°  ? 
£in  35°  or  cos  35°  ? 

4.  If  6  is  situated  between  0°  and  360°,  how  many  degrees  are  there  in 
6  if  tan  0  =  1?    Answer  the  similar  question  for  sin  0  =  %  ;  tan  $  =  —  1 . 

5.  Does  sin  60°  =  2  •  sin  30°  ?      Does  tan  60°  =  2  •  tan  30°  ?     What 
can  you  say  about  the  truth  of  the  equality  sin  2  0  =  2  sin  6  ? 

M)  The  Washington  Monument  is  555  ft.  high.  At  a  certain  place  in 
the  plane  of  its  base,  the  angle  of  elevation  of  the  top  is  60°.  How  far  is 
that  place  from  the  foot  and  from  the  top  of  the  tower  ? 

— "^.  A  boy  whose  eyes  are  5  ft.  from  the  ground  stands  200  ft.  from  a 
flagstaff.  From  his  eyes,  the  angle  of  elevation  of  the  top  is  30°.  How 
high  is  the  flagstaff  ? 


8.  A  tree  38  ft.  high  casts  a  shadow  38  ft.  long.  What  is  the  angle 
of  elevation  of/the  top  of  the  tree  as  seen  from  the  end  of  the  shadow  ? 
How  far  is  i*4rom  the  end  of  the  shadow  to  the  top  of  the  tree  ? 

i'rom  the  top  of  a  tower  100  ft.  high,  the  angle  of  depression  of 
two  stones,  which  are  in  a  direction  due  east  and  in  the  plane  of  the  base 
are  45°  and  30°  respectively.     How  far  apart  are  the  stones  ? 

.4ns.  100(  V3  -  1)  =  73.2  ft. 


18 


PLANE  TRIGONOMETRY 


[II,  §  15 


10.    Find  the  area  of  the  isosceles  triangle  in  which  the  equal  sides  10 
inches  in  length  include  an  angle  of  120°.  Ans.  25  V3  =  43.3  sq.  in. 

-^11.    Is  the  formula  sin  2  0  =  2  sin  0  cos  0  true  when  0  =  30°  ?     60°  ? 
120°? 

<l2!   From  a  figure  prove  that  sin  117°  =  cos  27°. 

13.    Determine  whether  each  of  the  following  formulas  is  true  when 
0  =  30°,  60°,  IHS0,  210D  : 

1  +  tan2  0  =  —  - — 

COS2  0 ' 

1  +  -1-  — *-, 

tan2  0      sin2  0 


sin2  0  -f  cos2  i 


1. 


,""i4.  Let  Pi(Xi,  ?/i)  and  P-z(x2,  yt)  be  any  two  points  the  distance  be- 
tween which  is  r  (the  units  on  the  axes  being  equal) .  If  0  is  the  angle 
that  the  line  PiP%  makes  with  the  x-axis,  prove  that 


x2  -  Xi    ,   ?/2 


*r=^»  =  2  r. 


}l6.  Computation  of  the  Value   of    One   Trigonometric 
Function  from  that  of  Another. 


J>±£Si 


Fig.  17 


Example  1.     Given  that  sin  0  =  f,  find  the 
values  of  the  other  functions. 

Since  sin  0  is  positive,  it  follows  that  0  is 
an  angle  in  the  first  or  in  the  second  quad- 
rant. Moreover,  since  the  value  of  the  sine 
is  |,  then  y  =  3  •  k  and  r  =  5  •  k,  where  k  is 
any  positive  constant  different  from  zero.  (Why?)  It  is,  of  course, 
immaterial  what  positive  value  we  assign  to  k,  so  we  shall  assign  the 
value  1.  We  know,  however,  that  the  abscissa,  ordinate,  and  distance 
are  connected  by  the  relation  x2  +  y2  =  r2,  and  hence  it  follows  that 
x  =  ±  4.  Figure  17  is  then  self-explanatory.  Hence  we  have,  for  the  first 
quadrant,  sin  0  =  f ,  cos  0  =  f ,  and  tan  0  =  £  ;  for  the  second  quadrant, 
sin  0  =  |,  cos  0  =  —  |,  tan  0  =  —  f . 


is  negative,  find  the  other  trigonometric  functions  of 
the  angle  0. 

Since  sin  0  is  positive  and  tan  0  is  negative,  0  must 
be  in  the  second  quadrant.  We  can,  therefore,  con- 
struct the  angle  (Fig.  18),  and  we  obtain  sin  0  =  ^T, 
cos  0  =  —  Y§,  tan  0  =  —  T\. 


Fig.  18 


II,  §  17] 


THE  RIGHT  TRIANGLE 


19 


k 


17.  Computation  for  Any  Angle.  Tables.  The  values  of 
the  trigonometric  functions  of  any  angle  may  be  computed  by 
the  graphic  method.  For 
example,  let  us  find  the 
trigonometric  functions  of 
35°.  We  first  construct 
on  square-ruled  paper, 
by  means  of  a  protractor, 
an  angle  of  35°  and  choose 
a  point  P  on  the  ter- 
minal line  so  that  OP 
shall  equal  100  units. 
Then  from  the  figure  we 
find  that  0^=82  units 
and  MP  =  57  units. 
Therefore 


TOT 

_L'. 

-; 

:: 

-l  >0~ 

TPT 

■':':': 

-[jjT 

-■■:■;■■ 

V* 

:::rj:::i 

b    ::: 

.ft-  ■•-■ 

■to — : — 

■i'>:.\.'..;. 
A     : 

'  • 

::  :i\ 

m 

V 

* 

10      XV      SO      40     SO      60       70     (SO      90     100 

Fig.  19 


sin  35°  =  tVv  =  °-57>  cos  35°  =  Tiro  =  °-82>  tan  35°  =  U  =  0.70. 

The  tangent  may  be  found  more  readily  if  we  start  by  tak- 
ing OA  =  100  units  and  then  measure  AB.  In  this  case, 
AB  =  70  units  and  hence  tan3o°  =  ^^  =  0.70. 

It  is  at  once  evident  that  the  graphic  method,  although 
simple,  gives  only  an  approximate  result.  However,  the  values 
of  these  functions  have  been  computed  accurately  by  methods 
beyond  the  scope  of  this  book.  The  results  have  been  put  in 
tabular  form  and  are  known  as  tables  of  natural  trigonometric 
functions.  Such  tables  and  how  to  use  them  will  be  discussed 
in  the  next  article. 

Figure  20  makes  it  possible  to  read  off  the  sine,  cosine,  or 
tangent  of  any  angle  between  0°  and  90°  with  a  fair  degree  of 
accuracy.  The  figure  is  self-explanatory.  In  reading  off 
values  of  the  tangent  use  the  vertical  line  through  100  for  angles 
up  to  55°,  and  the  line  through  10  for  angles  greater  than  55°. 
Its  use  is  illustrated  in  some  of  the  following  exercises. 


20 


PLANE  TRIGONOMETRY 


[II,  §  17 


10 

Fig 


so  to 

20.  —  Graphical,  T 


60  60  70  60  90  100 

able  oe  Trigonometric  Functions 


II,  §  18]  THE  RIGHT  TRIANGLE  21 

EXERCISES 

Find  the  other  trigonometric  functions  of  the  angle  6  when 
t£)tan0  =  -3.  3.    cos  0  =  1$.  5.   sin0  =  f. 

2.   sin0  =  -|.  4.    tan0=f  6.    cos0=  —  |. 

rl)  sin  0  =  f  and  cos  0  is  negative. 

8.  tan  0  =  2  and  sin  0  is  negative. 

9.  sin  0  =  —  \  and  tan  0  is  positive. 

10.  cos  0  =  §  and  tan  0  is  negative. 

11.  Can  0.6  and  0.8  be  the  sine  and  cosine,  respectively,  of  one  and 
the  same  angle  ?    Can  0.5  and  0.9  ?  Ans.  Yes  ;  no. 

12.  Is  there  an  angle  whose  sine  is  2  ?    Explain. 

13.  Determine  graphically  the  functions  of  20°,  38°,  70°,  110°. 

14.  From  Fig.  20,  find  values  of  the  following  : 
sin  10°,  cos  50°,  tan  40°,  sin  80°,  tan  70°,  cos  32°,  tan  14°,  sin  14°. 

15.  A  tower  stands  on  the  shore  of  a  river  200  ft.  wide.  The  angle  of 
elevation  of  the  top  of  the  tower  from  the  point  on  the  other  shore  exactly 
opposite  to  the  tower  is  such  that  its  sine  is  \.  Find  the  height  of  the 
tower. 

16.  From  a  ship's  masthead  160  feet  above  the  water  the  angle  of  de- 
pression of  a  boat  is  such  that  the  tangent  of  this  angle  is  /2.  Find  the 
distance  from  the  boat  to  the  ship.  Ans.    640  yards. 

18.  Use  of  Tables  of  Trigonometric  Functions.  Examina- 
tion of  the  tables  of  "  Four  Place  Trigonometric  Functions  " 
(p.  112)  shows  columns  headed  "  Degrees,"  "  Sine,"  "  Tangent," 
"  Cosine,"  and  under  each  of  the  last  three  named  a  column 
headed  "  Value  "  (none  of  the  other  columns  eoncern  us  at  pres- 
ent). Two  problems  regarding  the  use  of  these  tables  now 
present  themselves. 

1.    To  find  the  value  of  a  function  when  the  angle  is  given. 

(a)  Find  the  value  of  sin  15°  20'.  In  the  column  headed 
"  Degrees  "  locate  the  line  corresponding  to  15°  20'  (p.  113) ;  on 
the  same  line  in  the  "  value  "  column  for  the  "  Sine,"  we  read 
the  result :  sin  15°  20'  =  0.2644.  On  the  same  line,  by  using 
the  proper  column,  we  find  tan  15°  20'  =  0.2742,  and  cos  15°  20' 
=  0.9644. 


/  H 


22  PLANE  TRIGONOMETRY  [II,  §  IS 

(b)  Find  the  value  of  tan  57°  50'.  The  entries  in  the 
column  marked  "  Degrees "  at  the  top  only  go  as  far  as  45° 
(p.  116).  But  the  columns  marked  "  Degrees  "  at  the  bottom 
contain  entries  beginning  with  45°  (p.  116)  and  running  back- 
wards to  90°  (p.  112).  In  using  these  entries  we  must  use  the 
designations  at  the  bottom  of  the  columns.  Thus  on  the  line 
corresponding  to  57°  50'  (p.  115)  we  find  the  desired  value : 
tan  57°  50'  =  1.5900.  Also  sin  57°  50'  =  0.8465,  and  cos  57° 
50'  =  0.5324. 

(c)  Find  the  value  of  sin  34°  13'.  This  value  lies  between 
the  values  of  sin  34°  10'  and  34°  20'.     We  find  for  the  latter 

sin  34°  10'  =  0.5616 

sin  34°  20'  =  0.5640 

Difference  for  10'  =  0.0024 

Assuming  that  the  change  in  the  value  of  the  function 
throughout  this  small  interval  is  proportional  to  the  change  in 
the  value  of  the  angle,  we  conclude  that  the  change  for  1'  in  the 
angle  would  be  0.00024.  For  3',  the  change  in  the  value  of  the 
function  would  then  be  0.00072.  Neglecting  the  2  in  the  last 
place  (since  we  only  use  four  places  and  the  2  is  less  than  5), 
we  find  sin  34°  13'  =  0.5616  +  0.0007  =  0.5623.  This  process  is 
called  interpolation.  With  a  little  practice  all  the  work  in- 
volved can  and  should  be  done  mentally  ;  i.e.  after  locating  the 
place  in  the  table  (and  marking  it  with  a  finger),  we  observe 
that  the  "  tabular  difference  "  is  "  24  "  ;  we  calculate  mentally 
that  .3  of  24  is  7.2,  and  then  add  7  to  5616  as  we  write  down 
the  desired  value  0.5623. 

Similarly  we  find  tan  34°  13'  =  0.6800  (the  correction  to  be 
added  is  in  this  case  12.9  which  is  "  rounded  off  "  to  13)  and 
cos  34°  13'  =  0.8269.  (Observe  that  in  this  case  the  correction 
must  be  subtracted.     Why  ?) 

2.    To  find  the  angle  when  a   value  of  a  function  is  given. 

ere  we  proceed  in  the  opposite  direction.     Given  sin  A  = 


■J 


N 


II,  §  18]  THE  RIGHT  TRIANGLE  23 

0.3289 ;  find  A.  An  examination  of  the  sine  column  shows 
that  the  given  value  lies  between  sin  19°  10' (  =  0.3283)  and. 
>  sin  19°  20'(=  0.3311).  We  note  the  tabular  difference  to  be  28. 
The  correction  to  be  applied  to  19°  10'  is  then  fa  of  10'  =  f  f ' 
=  -1/'  =  2.1'.  Hence  A  =  19°  12.1'.  (With  a  four  place  table 
do  not  carry  your  interpolation  farther  than  the  nearest  tenth 
of  a  minute.)     (See  §  20.)    \ 

EXERCISES 

*    1.  For  practice  in  the  use  of  tables,  verify  the  following  : 

(a)  sin  18°  20' =  0.3145  (d)  sin  27°  14'  =0.4576  (g)  sin  62° 24M  =0.8862 
(6)  cos 37°  30' =0.7934  (e)  cos  34°  11' =0.8272  (h)  cos 59° 46' .2 =0.5034 
(c)  tan 75° 50' =3.9617  (/)  tan 68°  21' =  2.5173  (i)  tan  14° 55'.6 =0.2665 
Assume  first  that  the  angles  are  given  and  verify  the  values  of  the 
functions.  Then  assume  the  values  of  the  functions  to  be  given  and 
verify  the  angles. 

2.    A  certain  railroad  rises  6  inches  for  every  10  feet  of  track.     What 
angle  does  the  track  make  with  the  horizontal  ? 


NJ 


3.  On  opposite  shores  of  a  lake  are  two  flagstaffs  A  and  B.  Per- 
pendicular to  the  line  AB  and  along  one  shore,  a  line  BC  =  1200  ft.  is 
measured.  The  angle  ACB  is  observed  to  be  40°  20'.  Find  the  distance 
between  the  two  flagstaffs. 

4.  The  angle  of  ascent  of  a  road  is  8°.  If  a  man  walks  a  mile  up  the 
road,  how  many  feet  has  he  risen  ? 


\ 


5.  How  far  from  the  foot  of  a  tower  150  feet  high  must  an  observer, 
6  ft.  high,  stand  so  that  the  angle  of  elevation  of  its  top  may  be  23°. 5  ? 

6.  From  the  top  of  a  tower  the  angle  of  depression  of  a  stone  in  the 
lane  of  the.  base  is  40°  20'.     What  is  the  angle  of  depression  of  the  stone 

from  a  point  halfway  down  the  tower? 

7.  The  altitude  of  an  isosceles  triangle  is  24  feet  and  each  of  the  equal 
angles  contains  40°  20'.  Find  the  lengths  of  the  sides  and  area  of  the 
triangle. 

8.  A  flagstaff  21  feet  high  stands  on  the  top  of  a  cliff.  From  a  point 
on  the  level  with  the  base  of  the  cliff,  the  angles  of  elevation  of  the  top 
and  bottom  of  the  flagstaff  are  observed.  Denoting  these  angles  by  « 
and  /3  respectively,  find  the  height  of  the  cliff  in  case  sin  a  =  -/7  and 

Ans.   75  feet. 


\ 


24  PLANE  TRIGONOMETRY  [II,  §  18 

9.  A  man  wishes  to  find  the  height  of  a  tower  CB  which  stands  on  a 
horizontal  plane.  From  a  point  A  on  this  plane  he  finds  the  angle  of  ele- 
vation of  the  top  to  be  such  that  sin  CAB  =  f .  From  a  point  A'  which 
is  on  the  line  AC  and  100  feet  nearer  the  tower,  he  finds  the  angle  of 
elevation  of  the  top  to  be  such  that  tan  CA'B'=  §.  Find  the  height  of 
fche  tower. 

10.  Find  the  radius  of  the  inscribed  and  circumscribed  circle  of  a  regu- 
ar  pentagon  whose  side  is  14  feet. 

11.  If  a  chord  of  a  circle  is  two  thirds  of  the  radius,  how  large  an 
angle  at  the  center  does  the  chord  subtend  ? 


19.  Computation  with  Approximate  Data.  Significant 
Figures.  The  numerical  applications  of  trigonometry  (in  sur- 
veying, navigation,  engineering,  etc.)  are  concerned  with  com- 
puting the  values  of  certain  unknown  quantities  (distances, 
angles,  etc.)  from  known  data  which  are  secured  by  measure- 
ment. Now,  any  direct  measurement  is  necessarily  an  approxi- 
mation. A  measurement  may  be  made  with  greater  or  less 
accuracy  according  to  the  needs  of  the  problem  in  hand  —  but 
it  can  never  be  absolutely  exact.  Thus,  the  information  on  a 
signpost  that  a  certain  village  is  6  miles  distant  merely 
means  that  the  distance  is  6  miles  to  the  nearest  mile  —  i.e.  that 
the  distance  is  between  5±  and  6^  miles.  Measurements  in  a 
physical  or  engineering  laboratory  need  sometimes  to  be  made  to 
the  nearest  one '  thousandth  of  an  inch.  For  example  the  bore 
of  an  engine  cylinder  may  be  measured  to  be  3.496  in.,  which 
means  that  the  bore  is  between  3.4955  in.  and  3.4965  in. 

A  simple  convention  makes  it  possible  to  recognize  at  a 
glance  the  degree  of  accuracy  implied  by  a  number  represent- 
ing an  approximate  measure  (either  direct  or  computed).  This 
convention  consists  simply  in  the  agreement  to  write  no  more 
figures  than  the  accuracy  warrants.  Thus  in  arithmetic  6  and 
6.0  and  6.00  all  mean  the  same  thing.  This  is  not  so,  when 
these  numbers  are  used  to  express  the  result  of  measurement 
or  the  result  of  computation  from  approximate  data.  Thus  6 
means  that  the  result  is  accurate  to  the  nearest  unit,  6.0  that 


II,  §  20]  THE  RIGHT  TRIANGLE  25 

it  is  accurate  to  the  nearest  tenth  of  a  unit,  6.00  to  the  nearest 
hundredth  of  a  unit. 

These  considerations  have  an  important  bearing  on  practical 
computation.  If  the  side  of  a  square  is  measured  and  found 
to  he  3.6  in.  and  the  length  of  the  diagonal  is  computed  by 
the  formula :  diagonal  =/  side  x  V2^4t  would  be  wrong  to  write 
=  3.6  x  V2  =  3.6  x  1.4142  =  5.09112  in.  The  correct  result 
is  5.1  in.  For  the  computed  value  of  the  diagonal  cannot  be 
more  accurate  than  the  measured  value  of  the  side.  The  result 
5.09112  must  therefore  be  "  rounded  off "  to  two  significant 
figures,  which  gives  5.1.  As  a  matter  of  fact  for  the  purpose 
of  this  problem  V2  =  1.4142  should  be  rounded  before  multi- 
plication to  V2  =  1.4 ;  thereby  reducing  the  amount  of  labor 
necessary. 

A  number  is  "  rounded  off,"  by  dropping  one  or  more  digits 
at  the  right  and,  if  the  last  digit  dropped  is  5+,  6,  7,  8,  or  9 
increasing  the  preceding  digit  by  1.*  Thus  the  successive 
approximations  to  w  obtained  by  rounding  of  3.14159  •••  are 
3.1416,  3.142,  3.14,  3.1,  3. 

20.  The  Number  of  Significant  Figures  of  a  number  (in  the 
decimal  notation)  may  now  be  defined  as  the  total  number  of 
digits  in  the  number,  except  that  if  the  number  has  no  digits 
to  the  right  of  the  decimal  point,  any  zeros  occurring  between 
the  decimal  point  and  the  first  digit  different  from  zero  are 
not  counted  as  significant.  Thus,  34.06  and  3,406,000  are  both 
numbers  of  four  significant  figures :  while  3,406,000.0  is  a 
number  of  eight  significant  figures.! 

*  In  rounding  off  a  5  computers  round  off  to  an  even  digit.  Thus  1.415 
would  be  rounded  to  1.42,  whereas  1.445  would  be  rounded  to  1.44.  If  this 
rule  is  used  consistently  the  errors  made  will  tend  to  compensate  each  other. 

t  Confusion  will  arise  in  only  one  case.  For  example,  if  3999.7  were 
rounded  by  dropping  the  7  we  should  write  it  as  4000  which  according  to  the 
above  definition  would  have  only  1  significant  figure,  whereas  we  know  from 
the  way  it  was  obtained  that  all  four  figures  are  significant.  In  such  a  case 
we  may  underscore  the  zeros  to  indicate  they  are  significant  or  use  some 
other  device. 


26  PLANE  TRIGONOMETRY  [II,  §  20 

In  any  computation  involving  multiplication  or  division  the 
number  of  significant  figures  is  generally  used  as  a  measure  of 
the  accuracy  of  the  data.  A  computed  result  should  not  in 
general  contain  more  significant  figures  than  the  least  accurate 
of  the  data.  But  computers  generally  retain  one  additional 
figure  during  the  computation  and  then  properly  round  off  the 
final  result.  Even  then  the  last  digit  may  be  inaccurate  —  but 
that  is  unavoidable. 

The  following  general  rules  will  be  of  use  in  determining 
the  degree  of  accuracy  to  be  expected  and  in  avoiding  useless 
labor : 

1.  Distances  expressed  to  two  significant  figures  call  for 
angles  expressed  to  the  nearest  30'  and  vice  versa. 

2.  Distances  expressed  to  three  significant  figures  call  for 
angles  expressed  to  the  nearest  5',  and  vice  versa. 

3.  Distances  expressed  to  four  significant  figures  call  for 
angles  expressed  to  the  nearest  minute,  and  vice  versa. 

4.  Distances  expressed  to  five  significant  figures  call  for 
angles  expressed  to  the  nearest  tenth  of  a  minute,  and  vice, 
versa. 

In  working  numerical  problems  the  student  should  use  every 
safeguard  to  avoid  errors.  Neatness  and  systematic  arrange- 
ment of  the  work  are  important  in  this  connection.  All  work 
should  be  checked  in  one  or  more  of  the  following  ways. 
1.  Gross  errors  may  be  detected  by  habitually  asking  oneself : 
Is  this  result  reasonable  or  sensible  ?  2.  A  figure  drawn  to 
scale  makes  it  possible  to  measure  the  unknown  parts  and  to 
compare  the  results  of  such  measurements  with  the  computed 
results.  3.  An  accurate  check  can  often  be  secured  with  com- 
paratively little  additional  labor  by  computing  one  of  the 
quantities  from  two  different  formulas  or  by  verifying  a 
known  relation.  For  example,  if  the  legs  a,  b  of  a  right  tri- 
angle have  been  computed  by  the  formulas  a  =  c  sin  A  and 
b  =  c  cos  A,  we  may  check  by  verifying  the  relation  a2  +  b2  =  c2. 


II,  §  21] 


THE  RIGHT  TRIANGLE 


27 


Example.     A  straight  road  is  to  be  built  from  a  point  A  to  a  point  B 
which  is  5.92  miles  east  and  8.27  miles  north  of 
A.     What  will  be  the  direction  of  the  road  and 
its  length  ? 

5.92  ,  D       8.27 


Formulas : 
Therefore 


tan  A  = 


AB  = 


8  27  cos  A 

tan  A  =  0.716  and     A  =  35°  35', 
cos  ^  =  0.813  ^£  =  10.17.* 

Check  by    a2  +  &2  =  c2. 
From  a  table  of  squares  (p.  107,  see  §  21) 
(5.92)2=    35.05 

(8.27)2  =    68.39       (10.17)2  =  103.4. 
103.4 


21.  Use  of  Table  of  Squares.  Square  Roots.  The  table 
of  squares  of  numbers  (p.  106)  may  be  used  to  facilitate  com- 
putation. In  the  example  of  the  last  article,  we  required  the 
square  of  5.92.  We  find  5.9  on  p.  107  in  the  left-hand  column 
and  find  the  third  digit  2  at  the  head  of  a  certain  column.  At 
the  intersection  of  the  line  and  column  thus  determined  we 
find  the  desired  result  (5.92)2  =  35.05.  The  square  of  8.27  is 
found  similarly  at  the  intersection  of  the  line  corresponding 
to  8.2  and  the  column  headed  7.  To  find  (10.17)2,  we  find  the 
line  corresponding  to  1.0  (the  first  two  digits,  neglecting  the 
decimal  point)  and  find  (1.01)*  =  1.020  and  (1.02)2  =  1.040. 
By  interpolating,  as  explained  in  §  18,  we  find  (1.017)2  =  1.034. 
Now  shifting  the  decimal  point  one  place  in  the  "number" 
requires  a  corresponding  shift  of  two  places  in  the  square. 
Hence,  (10.17)*  =  103.4. 

The  table  can  also  be  used  to  find  the  square  root  of  a  num- 
ber. Thus  to  find  V2  we  find,  on  working  backwards  in  this 
table,  that  2  lies  between  1.988  [=(1.41)*]  and  2.016 [=(1.42)*]. 
By  interpolation  we  then  find  V2  m  1.414,  correct  to  four 
significant  places.  [Tabular  difference  =  28  ;  correction  =  -*^ 
=  4  in  the  fourth  place.] 


♦The  retention  of  four  significant  figures  in  AB  is  justified  because  the 
number  is  so  small  at  the  left. 


^ 


28  PLANE  TRIGONOMETRY  [II,  §  21 

EXERCISES 

1.  From  an  observing  station  357  ft.  above  the  water,  the  angle  of 
depression  of  a  ship  is  2°  15f.  Find  the  horizontal  distance  to  the  ship  in 
yards. 

2l  A  projectile  falls  in  a  straight  line  making  an  angle  of  25°  with  the 
horizontal.  Will  it  strike  the  top  of  a  tree  24  meters  high  which  is  72  meters 
from  the  point  where  the  projectile  would  strike  the  ground  ? 

3.   At  a  point  372  ft.  from  the  foot  of  a  cliff  surmounted  by  an  observa-  \*~*  T 
ion  tower  the  angle  of  elevation  of  the  top  of  the  tower  is  51°  25',  and  of  2-**}.^ 
the  foot  of  the  tower  31°  55'.     Find  the  height  of  the  cliff  and  of  the 
tower. 


f*.  How  far  from  the  foot  of  a  flagpole  130  ft.  high  must  an  observer 
stand  so  that  the  angle  of  elevation  of  the  top  of  the  pole  will  be  25°  ? 

5.  GA  is  a  horizontal  line,  T  is  a  point  vertically  above  i;  5a  point 

AG 

vertically  below  A.     The  angle  BG A  in  minutes  is Find  Z  BG  T 

4000 

in  degrees  and  minutes,  given  GA  =  10,340  meters  ;  AT  =  416.4  meters. 

6.  It  is  desired  to  find  the  height  of  a  wireless  tower  situated  on  the 
top  of  a  hill.  The  angle  subtended  by  the  tower  at  a  point  250  ft.  below" 
the  base  of  the  tower  and  at  a  distance  measured  horizontally  of  2830  ft. 
from  it  is  found  to  be  2°  42'.     Find  the  height  of  the  tower. 

7.  From  a  tower  428.3  ft.  high  the  angles  of  depression  of  two  objects 
tuated  in  the  same  horizontal  line  with  the  base  of  the  tower  and  on  the 

same  side  are  30°  22'  and  47°  37'.     Find  the  distance  between  them. 

8.  The  summit  of  a  mountain  known  to  be  13,260  ft.  high  is  seen  at 
an  angle  of  elevation  of  27°  12'  from  a  camp  located  at  an  altitude  of 
6359  ft.  Compute  the  air-line  distance  from  the  camp  to  the  summit  of 
the  mountain. 

9.  Two  towns  A  and  B,  of  which  B  is  25  miles  northeast  of  A,  are  to 
be  connected  by  a  new  road.  11  miles  of  the  road  is  constructed  from 
A  in  the  direction  N.  21°  E.  ;  what  must  be  length  and  direction  of  the 
remainder  of  the  road,  assuming  it  to  be  straight  ? 

22.  Applications  in  Navigation.  We  shall  confine  ourselves 
to  problems  interne  sailing;  i.e.  we  shall  assume  that  the  dis- 
tances considered  are  sufficiently  small  so  that  the  curvature  of 
the  earth  may  be  neglected. 


II,  §  22] 


THE  RIGHT  TRIANGLE 


29 


Definition.     The  course  of  a 

ship  is  the  direction  in  which  she 

is  sailing.     It  is  given  either  by 

the  points  of  a  mariner's  compass 

(Fig.  21)  as  K  E.  by  N.  or  in 

degrees    and   minutes  ■  measured 

clockwise  from  the  north.    Observe 

that  a   "  point "  on  a  mariner's 

compass   is   11°  15'.     Hence  for 

example,  the   course   of  a   ship 

could  be  given  either  as  N.  E.  by 

N.  or  as  33°  45;.     A  course  S.  E.  by  S.  is  the  same  as  a  course 

of  146°  15'. 

The  distance  a  ship  travels  on  a  given  course  is  always  given 
Departure  in  nautical  miles  or  knots.  A  knot  is  the  length 
of  a  minute  of  arc  on  the  earth's  equator.  (The 
earth's  circumference  is  then  360  x  60  =  21,600 
knots.)  The  horizontal  component  of  the  dis- 
tance is  called  the  departure,  the  vertical  com- 
ponent is  called  the  difference  in  latitude.  The 
departure  is  usually  given  in  miles  (knots),  the 
difference  in  latitude  in  degrees  and  minutes. 


Fig.  22 


Example.     A  ship  starts  from  a  position  in  22°  12'  N.  lati- 
tude, and  sails  321  knots  on  a  course  of  31°  15'.     Find  the 
difference  in  latitude,  the  departure,  and  the  latitude  of  the 
new  position  of  the  ship, 
diff.  in  lat.  =  distance  times  cosine  of  course 

=  321  cos  31°  15' 

=  321  x  0.855  =  274'  =  4°  34'. 
departure  =  distance  times  sine  of  course 

=  321  sin  31°  15' 

=  321  x  0.519  =  167  knots. 

Since  the  ship  is  sailing  on  a  course  which  increases  the  lati- 


30  PLANE  TRIGONOMETRY  [II,  §  22 

tude,  the  latitude  of  the  new  position  is  22°  12'  -f  4°  34'  =  26° 
46'  N. 

Knowing  the  difference  in  latitude  and  the  departure,  we  are 
able  to  calculate  the  new  position  of  the  ship,  if  the  original 
position  is  known.  In  the  preceding  example,  we  found  the 
latitude  of  the  new  position  from  the  difference  in  latitude. 
To  find  the  difference  in  longitude  from  the  departure  is  not 
quite  so  simple.  As  the  latitude  increases,  a  given  departure 
implies  an  increasing  difference  in  longitude.  Only  on  the 
equator  is  the  departure  of  one  nautical  mile  equivalent  to  a 
difference  in  longitude  of  one  minute. 

The  adjacent  figure  shows  a  departure  AB  in  latitude  <f>. 
The  difference  in  longitude  (in  minutes)  corresponding  to  AB 
is  clearly  the  number  of  nautical  miles  in 
CD.  Now  arcs  AB  and  CD  are  proportional 
to  their  radii  PA  and  OC.  Or, 

CD  =  °C~  .  AB  =  A**-.      (Why  ?) 
PA  cos<j>       v       J    J 

In  practice,  it  is  customary  to  take  for  <f> 
in  the  determination  of  difference  in  longi- 
tude the  so-called  middle  latitude,  i.e.  the 
latitude  halfway  between  the  original  latitude  and  the  final 
latitude. 

Thus  in  the  preceding  example,  the  original  latitude  was 
22°  12'  N,  the  final  latitude  was  26°  46'  N.  The  middle  lati- 
tude is  therefore  J  (22°  12'  +  26°  46')  =  24°  29'.     Hence 

,-pp  ,         ,    -,  departure 

difference  in  longitude  = . . , ,, — , — ■, — =r- 

cosme  or  middle  latitude 

167  167   =  lg4,  m  30  4, 


Fig.  23 


cos  24°  29'      0.910 


The  determination  of  the  position  of  a  ship  from  its  course 
and  distance  is  known  as  dead  reckoning.  It  is  subject  to  con- 
siderable inaccuracy  and  must  often  in  practice  be  checked  by 


II,  §  22]  THE  RIGHT  TRIANGLE  31 

direct  determination  of  position  by  observations  on  the  sun 
or  stars. 

EXERCISES 

1.  A  ship  sails  N.  E.  by  E.  at  the  rate  of  12  knots  per  hour.  Find  the 
rate  at  which  it  is  moving  north. 

2.  A  ship  sails  N.  E.  by  N.  a  distance  of  578  miles.  Find  its  departure 
and  difference  in  latitude. 

3.  A  ship  sails  on  a  course  of  73°  until  its  departure  is  315  miles.  Find 
the  actual  distance  sailed.     Find  also  its  difference  in  latitude. 

4.  A  ship  sails  from  latitude  47°  \&  N.  670  miles  on  a  course  N.  W. 
by  N.     Find  the  latitude  arrived  at. 

5.  A  ship  sails  from  latitude  30°  24'  N.  and  after  25  hours  reaches  lati- 
tude 35°  26'  N.  Its  course  was  N.  N.  W.  Find  the  average  speed  of  the 
ship. 

6.  A  vessel  sails  from  lat.  24°  30'  N.,  long.  30°  15  W.,  a  distance  of  692 
miles  on  a  course  of  32°  20'.  Find  the  latitude  and  longitude  of  its  new 
position. 

7.  A  vessel  sails  from  lat.  10°  30'  S.,  long.  167°  20'  W.,  a  distance  of 
692  miles  on  a  course  of  152°  30f.  Find  the  latitude  and  longitude  of  its 
new  position. 


CHAPTEE   III 

SIMPLE  TRIGONOMETRIC  RELATIONS 

/2Z.   Other  Trigonometric  Functions.     The  reciprocals  of 
'  the  sine,  the  cosine,  and  the  tangent  of  any  angle  are  called, 
respectively,  the  cosecant,  the  secant,  and  the  cotangent  of 
that  angle.     Thus, 

cosecant  0  =  dlstance  of  P=  -   (provided  y  =#=  0). 
ordinate  of  P     y 

,  r.      distance  of  P     r    ,         . ,    q       ,  AN 

secant  0  =  — — : =  -   (provided  x^=0). 

abscissa  of  P     x 

f\  nsoissj-i  Or   r^       ^v 

cotangent  0  = : — —  —  -   (provided  y  ^=  0). 

ordinate  of  P     p 

These  functions  are  written  esc  0,  sec  0,  ctn  0.     From  the 
definitions  follow  directly  the  relations 


esc  6=  — ,   sec  8  = -,   ctn  6 


sin  0 '  cos  9  '  tan  8 

or 

esc  0  •  sin  0  =  1,  sec  6  •  cos  0  =  1,  ctn  0  •  .tan  0  =  1. 

To  the  above  functions  may  be  added  versed  sine  (written  versin),  the 
co versed  sine  (written  coversin),  and  the  external  secant  (written  exsec), 
which  are  defined  by  the  equations  versin  0  =  1  —  cos  0,  coversin  0  = 
1  —  sin  0,  and  exsec  0  =  sec  0  —  1.  Of  importance  in  navigation  and  service- 
able in  other  applications  (see  §  88)  is  the  haversine  (written  hav) 
which  is  defined  to  be  equal  to  one  half  the  versed  sine  ;  i.e. 

have  =  |(1  — cos  0). 

r24.  The  Representation  of  the  Functions  by  Lines.  Con- 
sider an  angle  0  in  each  quadrant  and  about  the  origin  draw 

32 


Ill,  §  24]     SIMPLE  TRIGONOMETRIC  RELATIONS        33 


Fig.  24 

a  circle  of  unit  radius.  Let  P(x,  y)  be  the  point  where  the 
circle  meets  the  terminal  side  of  6.     Then 

sin  6  =  ¥=zy,    cos  6  =  ^  =  x, 

i.e.  the  sine  is  represented  by  the  ordinate  of  P  and  the  cosine 
by  the  abscissa.  Hence  the  sine  and  cosine  have  respectively 
the  same  signs  as  the  ordinate  and  abscissa  of  P. 

If  we  draw  a  tangent  to  the  circle  at  the  point  A  where  the 


Fig.  25 


circle  meets  the  a^axis  and  let  the  terminal  line  of  9  meet  this 
tangent  in  Q,  we  have 

tenO  =  ^Q  =  AQ,    sec0  =  -^2=OQ. 

Note  that  when  6  =  90°,  270°,  and  in  general  90  +  n  .  360°, 
270°  +  n  •  360°,  where  n  is  any  integer,  there  is  no  length  AQ 
cut  off  on  the  tangent  line  and  hence  these  angles  have  no 
tangents. 

If  we  draw  a  line  tangent  to  the  circle  at  the  point  B  where 

D 


34 


PLANE  TRIGONOMETRY 


[HI,  §  24 


the  circle  cuts  the  y-axis  and  let  the  terminal  line  of  6  cut 
this  tangent  in  B,  we  have 

ctn0=zBK=B^  and  csc £  =  OR  =  QR 


Fig.  26 


EXERCISES 

1.  From  Fig.  24  prove  sin2  0  +  cos2  0  =  1. 

2.  From  Fig.  25  prove  1  +  tan2  0  =  sec2  0. 
I.'  From  Fig.  26  prove  1  +  ctn2  0  as  csc2  0. 


It  Colon. 

A^ 

B*N 

/\\ 

/    ^ 

1    eK 

^ 

\^ 

25.  Relations  among  the  Trigonometric  Functions.    As 

one  might  imagine,  the  six  trigonometric  functions  sine,  cosine, 
tangent,  cosecant,  secant,  cotangent  are  connected  by  certain 
relations.     We  shall  now  find  some  of  these  relations. 

From  Fig.  9  (§  10)  it  is  seen  that  for  all  cases  we  have 
(1)  x2  +  y2  =  r2. 

If  we  divide  both  sides  of  (1)  by  r2,  we  have 


+ 


v- 


or 


sin2  6  +  cos2  6  =  1. 

Dividing  both  sides  of  (1)  by  x2,  we  have 


1     (by  hypothesis  r  =£  0) ; 


1+%=b  (if**°>- 


Therefore, 


1  +  tan2  6  =  sec2  6. 

Similarly  dividing  both  sides  of  (1)  by  y2  gives 


or 


r 

ctn2  6  +  1 


+  !  =  -,     (i*y*0); 


r 

csc2  6. 


Ill,  §  26]     SIMPLE  TRIGONOMETRIC  RELATIONS        35 

Moreover,  we  have 

y 

tane  =  ^  =  :=^° 
x     x     cos  8 

i  r 

and,  similarly, 

•     cos  6 
ctn6  =  ^— S-. 
,     sin  8 

26.  Identities.  By  means  of  the  relations  just  proved 
any  expression  containing  trigonometric  functions  may  be 
put  into  a  number  of  different  forms.  It  is  often  of  the 
greatest  importance  to  notice  that  two  expressions,  although 
of  a  different  form,  are  nevertheless  identical  in  value.  (How 
was  an  "  identity"  defined  in  algebra  ?) 

The  truth  of  an  identity  is  usually  established  by  reducing 
both  sides,  either  to  the  same  expression,  or  to  two  expres- 
sions which  we  know  to  be  identical.  The  following  examples 
will  illustrate  the  methods  used. 

Example  1.     Prove  the  relation  sec2  0  +  esc2  0  =  sec2  0  esc2  0. 
We  may  write  the  given  equation  in  the  form 

+  -^—  =  sec2  0  esc2  0, 


cos2  0     sin2  0 

sin2  0  +  cos2  0 
cos2  0  sin2  0 

1 


=  sec2  0  esc2  0, 
=  sec2  0  esc2  0, 


which  reduces  to 


cos2  0  sin2  0 

sec2  0  esc2  0  =  sec2  0  esc2  0. 


Since  this  is  an  identity,  it  follows,  by  retracing  the  steps,  that  the 
given  equality  is  identically  true. 

Both  members  of  the  given  equality  are  undefined  for  the  angles  0°,  90°, 

180°,  270°,  360°,  or  any  multiples  of  these  angles. 

cos2  0 

Example  2.     Prove  the  identity  1  4-  sin  0  — 

J  1  -  sin  0 

Since  cos2  0  =  1  —  sin2  0,  we  may  write  the  given  equation  in  the  form 

1  +  sin  0  =  1  ""  S1"2  9  or  1  +  sin  0  =  1  +  sin  0. 
1  -  sin  0 


36  PLANE  TRIGONOMETRY  [III,  §  26 

As  in  Example  1,  this  shows  that  the  given  equality  is  identically  true. 

The  right-hand  member  has  no  meaning  when  sin  0  =  1 ,  while  the  left- 
hand  member  is  defined  for  all  angles.  We  have,  therefore,  proved  that 
the  two  members  are  equal  except  for  the  angle  90°  or  (4  n-f  1)90°,  where 
n  is  any  integer. 

The  formulas  of  §  25  may  be  used  to  solve  examples  of  the 
type  given  in  §  16. 

Example  3.  Given  that  sin  0  =  ft  and  that  tan  0  is  negative,  find  the 
values  of  the  other  trigonometric  functions. 

Since  sin2  0  +  cos2  0  =  1,  it  follows  that  cos  0  =  ±  Jf ,  but  since  tan  0  is 
negative,  0  lies  in  the  second  quadrant  and  cos0  must  be  —  ||.  More- 
over, the  relation  tan  0  =  sin  0/cos  0  gives  tan  0  =  —  ft.  The  reciprocals 
of  these  functions  give  sec  0  =  —  ||,  esc  0  =  y,  ctn  0  —  —  *g. 

EXERCISES 

1.  Define  secant  of  an  angle  ;  cosecant ;  cotangent. 

2.  Are  there  any  angles  for  which  the  secant  is  undefined  ?  If  so, 
what  are  the  angles  ?  Answer  the  same  question  for  cosecant  and  co- 
tangent. 

3.  Define  versed  sine  ;  co versed  sine  ;  haversine. 

4.  Complete  the  following  formulas  : 

sin2  6  +  cos2  0  =  ?     1  +  tan2  0  =  ?     1  +  ctn2  0  =  ?    tan  0  =  ? 
Do  these  formulas  hold  for  all  angles  ? 

5.  In  what  quadrants  is  the  secant  positive  ?  negative  ?  the  cosecant 
positive  ?  negative  ?  cotangent  positive  ?  negative  ? 

6.  Is  there  an  angle  whose  tangent  is  positive  and  whose  cotangent  is 
negative  ? 

7.  In  what  quadrant  is  an  angle  situated  if  we  know  that 

(a)  its  sine  is  positive  and  its  cotangent  is  negative  ? 

(b)  its  tangent  is  negative  and  its  secant  is  positive  ? 

(c)  its  cotangent  is  positive  and  its  cosecant  is  negative  ? 

— ' — *«••«£•    Express  sin2  0  +  cos  0  so  that  it  shall   contain  no  trigonometric 
sA  function  except  cos  0. 

9.    Transform  (1  +  ctn2  0)csc  0  so  that  it  shall  contain  only  sin  0. 

10.  Which  of  the  trigonometric  functions  are  never  less  than  one  in 
absolute  value  ? 

11.  For  what  angles  is  the  following  equation  true  :  tan  0  =  ctn  0  ? 
"^^wU-        12.    How  many  degrees  are  there  in  0  when  ctn  0  =  1?     ctn  0  —  —  1  ? 

sec  0  =  V2  ?     esc  0  =  V£  ? 

(  H  <^c3? 


Ill,  §  27]     SIMPLE  TRIGONOMETRIC  RELATIONS        37 


13.  Determine  from  a  figure  the  values  of  the  secant,  cosecant,  and 
cotangent  of  30°,  150°,  210°,  330°. 

14.  Determine  from  a  figure  the  values  of  the  secant,  cosecant,  and 
cotangent  of  45°,  135°,  225°,  315°. 

15.  Determine  from  a  figure  the  values  of  the  sine,  cosine,  tangent, 
secant,  cosecant,  and  cotangent  of  60°,  120°,  240°,  300°. 

16.  Find  0  from  the  following  equations. 

(a)  sin0=£.  (i)  tan0=  —  1. 

(6)  sin  0  =  -  \.  (j)  ctn  0  =  -  1. 

(c)  cos  0  =  \.  (k)  tan  0  =  1. 

(d)  cos  0  =  -  £.  (I)  ctn  0  =  1. 

(e)  sec  0  =  2.  (m)  tan2  0  =  3. 
(/)  sec  0  =  -  2.  (n)  sin  0  =  0. 
(gr)  esc  0  =  2.  (b)  cos  0  =  0. 
(h)  esc  0  =•-  2.  O)  tan  0  =  0. 

Prove  the  following  identities  and  state  for  each  the  exceptional  values 
of  thjt variables,  if  any,  for  which  one  or  both  members  are  undefined  : 


cos 

0  tan  0  = 

sin0. 

sin 

0  ctn  0  = 

COS0. 

14 

sin0 

COS0 

tor*  (  H/»*—    •  C^r 

cos  0         1  —  sin  0 
sin2  0  —  cos2  0  =  2  sin2  0-1. 
(1  —  sin2  0)csc2  0  =  ctn2  0. 
tan  0  +  ctn  0  =  sec  0  esc  0. 
[x  sin  0  +  y  cos  0]2  4-  [x  cos  0  —  y  sin  0]2  =  x2  -f  y2. 

2^ =Cos0. 

tan  0  +  ctn  0 

1  —  ctn4  0  =  2  esc2  0  -  esc4  0. 

26.  tan2  0  -  sin2  0  =  tan2  0  sin?  0. 

27.  2(1  +  sin  0)  (1  +  cos  0)  =  (1  +  sin  0  +  cos  0)2 

28.  sin6  0  +  cos«  0=1-3  sin2  0  cos2  0 
esc  0  esc  0 


1- 

27.   The  Trigonometric  Functions  of  90° -0.    Figure  21 
represents  angles  6  antf  90°  —  0,  when  0  is  in  each  of  the  four 


X03 


-(vr-  ^A-v^ 


/ 


(ooo  -V  y\>^~ 


uuSe*^ 


38 


PLANE  TRIGONOMETRY 


[III,  §  27 


quadrants.     Let  OP  be  the  terminal  line  of  0  and   OP'  the 
terminal  line  of  90°  -  0.     Take   OP'  =  OP  and  let  (x,  y)  be 


Fig.  27 


the  coordinates  of  P  and  (x',  y')  the  coordinates  of  P\ 
in  all  four  figures  we  have 

x'  =  y>  yf  =  x>  r'  =  r. 


Then 


Hence 


sin(9O°-0)  =  ^  =  -: 


cos 


Also, 


cos  (90°  -  0)  =  -  =  2  =  sin  0, 
r      r 

tan (90°  -6)  =  ^  =  -=ctn0. 
x'      y 


esc  (90°  —  0)=sec0, 
sec  (90° -0)=  esc  0, 
ctn  (90° -0)=  tan  0. 

Definition.  The  sine  and  cosine,  the  tangent  and  cotangent, 
the  secant  and  cosecant,  are  called  co-functions  of  each  other. 

The  above  results  may  be  stated  as  follows  :  Any  function 
of  an  angle  is  equal  to  the  corresponding  co-function  of  the  com- 
plementary angle.* 

28.   The  Trigonometric  Functions  of  180°  —  6.     By  draw- 
ing figures  as  in  §  27,  the  following  relations  may  be  proved : 
sin  (180°  -  6)  =  sin  0,  esc  (180°  -  6)  =  esc  0, 

cos  (180°  -  0)  =  -  cos  0,  sec  (180°  -  0)  =  -  sec  0, 

tan  (180°  -  0)  =  -  tan  0,  ctn  (180°  -  0)  =  -  ctn  0. 

The  proof  is  left  as  an  exercise. 

*  Two  angles  are  said  to  be  complementary  if  their  sum  is  90°,  regardless 
of  the  size  of  the  angles. 


Ill,  §  29]      SIMPLE  TRIGONOMETRIC   RELATIONS        39 

29.  The  result  of  §  27  shows  why  it  is  possible  to  arrange 
the  tables  of  the  trigonometric  functions  with  angles  from  0° 
to  45°  at  the  top  of  the  pages  and  angles  from  45°  to  90°  at 
the  bottom  of  the  pages.  For  example,  since  sin  (90° — 0)  =  cos  0, 
the  entry  for  cos  0  will  serve  equally  well  for  sin  (90°  —  6). 
As  particular  instances  we  may  note  sin  67°  =  cos  23°,  tan  67° 
=  ctn  23°,  cos  67°  =  sin  23°.     Verify  these  from  the  table. 

The  result  of  §  28  enables  us  to  find  the  values  of  the  func- 
tions of  an  obtuse  angle  from  tables  that  give  the  values  only 
for  acute  angles.  It  will  be  noted  that  §  28  says  that  any 
function  of  an  obtuse  angle  is  in  absolute  value  equal  to  the  same 
function  of  its  supplementary  angle  but  may  differ  from  it  in 
sign. 

Thus  to  find  tan  137°  we  know  that  it  is  in  absolute  value 
the  same  as  tan  (180°  -  137°)  =  tan  43°  =  0.9325.  But  tan  137° 
is  negative.     Hence 

tan  137°  =  -  0.9325. 

Similarly,  sin  137°=      0.6820. 

cos  137°  =  -  0.7314. 

EXERCISES 
Find  the  values  of  the  following  : 

tan  146°,   sin  136°,   cos  173°,   tan  100°,    cos  96°,     sin  138°, 
tan  98°,     sin  145°,    cos  168°,    cos  138°,   tan  173°,    cos  157°. 


CHAPTER   IV 


OBLIQUE  TRIANGLES 

30.   Law  of    Sines.     Consider  any  triangle  ABC  with  the 
altitude  CD  drawn  from  the  vertex  C  (Fig.  28). 


In  all  cases  we  have  sin  A 


Therefore,  dividing,  we  obtain 

sin  A      a  a 

=  - ,  or 

sin  B      b  sin  A 


(i) 


(2) 


sin  B 

If  the  perpendicular  were  dropped  from  B,  the  same  argu- 
ment would  give  a/sin  A  =  c/sin  C.     Hence,  we  have 
a  b  c 

sin  A      sin  B      sin  C 

This  law  is  known  as  the  law  of  sines  and  may  be  stated  as 
follows :  Any  two  sides  of  a  triangle  are  proportional  to  the 
sines  of  the  angles  opposite  these  sides. 

31.   Law  of  Cosines.     Consider  any  triangle  ABC  with  the 
altitude  CD  drawn  from  the  vertex  C  (Fig.  29). 
In  Fig.  29  a 

AD  =  b  cos  A  ;  CD  =  b  sin  A  ;  DB  =  c  —  b  cos  A. 
In  Fig.  29  b 

AD  =  —  b  cos  A ;  CD  =  b  sin  A  ;  DB  =  c  —  b  cos  A. 
In  both  figures 

a2  =  DB2  -f  CZ)2. 


40 


d 


IV,  §  32] 
Therefore 


OBLIQUE  TRIANGLES 


41 


a1  =  c2  -  2  be  cos  A  +  b2  cos2  A  +  b2  sin2  A 
=  c2  —  2bc  cos  ^  +  (cos2  A  +  sin2  ^1)62, 

o  c 


whence 

a2  _  tf.  +  C2  _  2  be  cos  i4. 

The  result  holds  also  when  A  is  a  right  angle.     Why  ? 
Similarly  it  may  be  shown  that 

b2  =  c2  +  a2  —  2  ca  cos  £, 

c2  =  a2  +  b2  —  2  a6  cos  C. 

Any  one  of  these  similar  results  is  called  the  law  of  cosines. 
It  may  be  stated  as  follows : 

Tlie  square  of  any  side  of  a  triangle  is  equal  to  the  sum  of  the 
squares  of  the  other  two  sides  diminished  by  twice  the  product  of 
these  two  sides  times  the  cosine  of  their  included  angle* 

32.    Solution  of  Triangles.     To  solve  a  triangle  is  to  find 

the  parts  not  given,  when  certain   parts  are   given.      From 

geometry  we  know  that  a  triangle  is  in  general  determined 

when   three   parts   of   the   triangle,  one  of  which   is  a  side, 

are    given.f     Eight    triangles     have     already    been     solved 

(§  15),  and  we  shall  now  make  use  of  the  laws  of  sines  and 

cosines    to  solve  oblique  triangles.     The    methods   employed 

will   be   illustrated   by   some    examples.     It    will    be    found 

advantageous  to  construct  the  triangle  to  scale,  for  by  so  doing 

one  can  often  detect  errors  which  may  have  been  made. 

*  Of  what  three  theorems  in  elementary  geometry  is  this  the  equivalent  ? 
t  When  two  sides  and  an  angle  opposite  one  of  them  are  given,  the  triangle 
is  not  always  determined.    Why  ? 


42  PLANE  TRIGONOMETRY  [IV,  §  33 

33.   Illustrative  Examples. 

Example  1.     Solve  the  triangle  AB  C,  given 
=  276    A  =  30°  20',  B  =  60°  45',  a  =  276. 

Solution  : 

C  =  180°  -  (A+  B)  =  180°  -  91° 5'  =  88° 55'; 

:  _  a  sin  B  _  270  sin  00°  45'  =  (270)  (0*8725)  =  476  9  . 
sin^l   '      sin  30°  20'  0.5050 


also 


c  -  ^_sill_^  -  276  sin  88°  55'  _  (276)  (0.9998)  __  546  4 
sin  A  sin  30°  20'  0.5050 


"Check  :  It  is  left  as  an  exercise  to  show  that  for  these  values  we  have 
c2  =  a2  +  b2  —  2  ab  cos  C. 

Example  2.     Solve  the  triangle  ABC,    given 
A  =  30°,  b  =  10,  a  =  6. 

{?  Constructing  the   triangle   ABC,   we  see  that 

two  triangles  ABX  C  and  AB2C  answer  the  descrip- 
*   tion  since  b  >  a  >  altitude  CD. 

Solution  :  Now 

***!  =  *,  or  sin  B,  =  ^^  =0.833, 

sin  A       a  a 

whence  B\  =  56°.  5. 

But 

B2  =  180°  -  Bx  =  180°  -  56°.5  =  123°.5, 
and 
Ci  =  180°  -{A  +  50=  180°  -  86°.5  =  93°. 5, 
C2  =  180°  -  {A  +  ft)  =  180°  -  153°. 5  =  26u.5. 
Now 

C2  _  sin  C2      or    C2  _  a  sin  C2  _  (6)  (0.446)  _  g  35 
a      sin  ^1  '  sin  -4  0.500 

Also 

Ci  =  8inC1.    or    Ci=asinC2==(6)(0.998)_1198 
a      sin  -4  '  sin  J.  0.500 

Check:  Ci2  =  a2  +  &2  —  2  ab  cos  0i. 

143.5  =  36  +  100  +(2)  (6)  (10)  (0.061)  =  143.3. 

C22  =  a2  +  ^2  _  2  a&  cos  C2. 
28.62  =  36  +  100-(2)(6)(10)(0.895)  =  28.60. 


IV,  §  33] 


OBLIQUE  TRIANGLES 


43 


Example  3.     Solve  the  triangle  ABC,  given  a  =  10,  6  =  6,  C  =  40°. 

Solution  :    c2  =  a2  +  62  —  2  ab  cos  (7 

=  100  +  36  -  (120) (0.766)=  44.08. 
Therefore    c  =  6.64.     Now 

sin^  =  asinC=(10)(0.643)  = 

c  6.64  ' 

i.e.  A  =  104°. 5.     Likewise, 

sing  =  6sinC=  (6X0.643)  = 

c  6.64  ' 


Check  :  A  +  B  +  C  =  180°.0. 

Example  4.     Solve   the   triangle   ABC  when 
C  a  =  7,  6  =  3,  c  =  5. 

From  the  law  of  cosines, 


&2   i   C2  _  a2  i 

COSA=        26c        =-,  =  -0.800, 

cos  B  =  ?l±^l»!  =  15  =  0.928, 
2  ac  14 

coSC  =  ?l+-^li?  =  11  =  0.786. 
2  06  14 


i.e. 

B 

=  35° 

.5. 

s* 

=  S 

JS* 

a  = 
Fig. 

7 
33 

Therefore 


.4  =  120°,  Z*  =  21°.8,  C  =  38°.2. 
Check  :    A  +  B  +  C  =  180°.0 

EXERCISES 


\£)  Solve  the  triangle  ABC,  given 

/4  (a)  ^1  =  a0°,         B  =  70°, 

a  =  100  ; 

(6)  A  =  40°,         B  =  70°, 

c  =  110; 

xj\c)  A  =  45°.5,      <7  =  68°.5, 

6  =  40; 

\d)  B=60°.5,      C  =  44°20', 

c  =  20; 

e)Va  =  30,     &  =  54,     C  =  50°  ; 

^  a  =  10, 

6  =  12,     c  =  14  ; 

2Q  6  =  8,       a  =  10,     C  =  60°  ; 

Jtf)   a  =  21, 

6  =  24,     c  =  28. 

2.    Determine  the  number  of  solutions  of  the  triangle  ABC  when 


(a)  A  =  30°,  6  =  100,  a  =    70 

(6)  4  =  30°,  6  =  100,  a  =  100 

(c)  21  =  30°,  6  =  100,  a  =    50 

(d)  4  =  30°,  6  =  100,  a  =    40 


(e)  A  =  30°,  6  =  100,  a  =  120  ; 
(/)  J.  =  106°,  6  =  120,  a  =  16  ; 
(gr)  4=   90°,  6=    15,  a  =    14. 


44 


PLANE  TRIGONOMETRY 


[IV,  33 


3.    Solve  the  triangle  ABC  when 
(a)  A  =  37°  20',    a  =  20,    6  =  26  ;       (c)   4  =  30°,     a  =  22,     6  =  34. 
(6;   ^L  =  37°  20',    a  =  40,    6  =  26; 

(4/^h  order  to  find  the  distance  from  a  point  A  to  a  point  B,  a  line 
-4C  and  the  angles  CAB  and  .A  (72?  were  measured  and  found  to  be 
300  yd.,  60°  30',  5.6°  10'  respectively.     Find  the  distance  AB. 

5.  In  a  parallelogram  one  side  is  40  and  one  diagonal  90.  The  angle 
between  the  diagonals  (opposite  the  side  40)  is  25°.  Find  the  length  of 
the  other  diagonal  and  the  other  side.     How  many  solutions  ? 

6.  Two  observers  4  miles  apart,  facing  each  other,  find  that  the  angles 
of  elevation  of  a  balloon  in  the  same  vertical  plane  with  themselves  are 
60°  and  40°  respectively.  Find  the  distance  from  the  balloon  to  each 
observer  and  the  height  of  the  balloon. 

7.  Two  stakes  A  and  B  are  on  opposite  sides  of  a  stream  ;  a  third 
stake  C  is  set  100  feet  from  A,  and  the  angles  A  C B  and  CAB  are  observed 
to  be  40°  and  110°,  respectively.     How  far  is  it  from  A  to  B  ? 

8.  The  angle  between  the  directions  of  two  forces  is  60°.  One  force 
is  10  pounds  and  the  resultant  of  the  two  forces  is  15  pounds.  Find  the 
other  force.* 

9.  Eesolve  a  force  of  90  pounds  into  two  equal  components  whose 
directions  make  an  angle  of  60°  with  each  other. 

10.  An  object  B  is  wholly  inaccessible  and  invisible  from  a  certain 
point  A.  However,  two  points  C  and  D  on  a  line  with  A  may  be  found 
such  that  from  these  points  B  is  visible.  If  it  is  found  that  CD  =  300  feet, 
AC  =  120  feet,  angle  DCB  =  70°,  angle  CDB  -  50°,  find  the  length  AB. 

11.  Given  a,  6,  A,  in  the  triangle  ABC.  Show  that  the  number  of 
possible  solutions  are  as  follows  :  0 

A<90° 

f  a  <  b  sin  A         no  solution, 
I  b  sin  A  <  a  <  b  two  solutions, 


a>b 


one  solution. 


|  a  =  b  sin  A  j 
^^90° 

(a_6     no  solution, 
a  >  b     one  solution. 
12.   The  diagonals  of  a  parallelogram  are  14  and  16  and  form  an  angle 
of  50°.     Find  the  length  of  the  sides. 

*  It  is  shown  in  physics  that  if  the  line  segments  AB 
and  AC  represent  in  magnitude  and  direction  two 
forces  acting  at  a  point  A,  then  the  diagonal  AD  of  the 
parallelogram  ABCD  represents  both  in  magnitude  and 
direction  the  resultant  of  the  two  given  forces. 


IV,  §  34] 


OBLIQUE  TRIANGLES 


45 


13.  Resolve  a  force  of  magnitude  150  into  two  components  of  100  and 
80  and  find  the  angle  between  these  components. 

14.  It  is  sometimes  desirable  in  surveying  to  extend  a  line  such  as  AB 


in  the  adjoining  figure.     Show  that  this  can  be  done  by  means  of  the 
broken  line  ABCDE.     What  measurements  are  necessary  ? 

15.  Three  circles  of  radii  2,  6,  5  are  mutually  tangent.  Find  the  angles 
between  their  lines  of  centers. 

16.  In  order  to  find  the  distance  between  two  objects  A  and  B  on  op- 
posite sides  of  a  house,  a  station  C  was  chosen,  and  the  distances  CA 
=  500  ft.,  CB  =  200  ft.,  together  with  the  angle  ACB  =  65° 30',  were 
measured.     Find  the  distance  from  A  to  B. 

17.  The  sides  of  a  field  are  10,  8,  and  12 
rods  respectively.  Find  the  angle  opposite  the 
longer  side. 

18.  From  a  tower  80  feet  high,  two  objects, 
A  and  B,  in  the  plane  of  the  base  are  found  to 
have  angles  of  depression  of  13°  and  10°  respec- 
tively ;  thejiorizontal  angle  subtended  by  A  and  B  at  the  foot  C  of  the 
tower jedBP.     Find  the  distance  from  A  to  B. 


Areas  of  Oblique  Triangles. 

When  tivo  sides  andjjie  included  angle  are  given. 
noting  the  area  byQfjJire  have  from  geometry 

8  =  i  ch, 
but  h  =  b  sin  A  ;  therefore 
(1)  S  =  ±cbsmA. 

Likewise, 

S  =  i  ab  sin  C  and  S  =  \ac  sin  B. 


Fig.  &i 


2.    When  a  side  and  two  adjacent  angles  are  given. 

Suppose  the  side  a  and  the  adjacent  angles  B  and  C  to  be 
given.  We  have  just  seen  that  8  =  \  ac  sin  B.  But  from  the 
law  of  sines  we  have 

a  sin  C 


sin  A 


46  PLANE  TRIGONOMETRY  [IV,  §  34 

Therefore 

Q_a2 '  sin  B  ♦  sin  C 
2  sin^t 

But  sin  A  =  sin  [180°  -  (B  +  C)]  =  sin  (5  +  C).     Therefore 

q  _  a2  sin  jB  sin  (7 
^~  ~2  sin (B+C)' 

j    3.  jWhen  the  three  sides  are  given. 

^*"W e  have  seen  that  S  =  \  be  sin  A.     Squaring  both  sides  of 
this  formula  and  transforming,  we  have 

£2  =  2_1  sin2^l  =  — (l-cos2^l) 
4  4 

=  1(1  +  003.4).  |(1- cos^); 

whence,  by  the  law  of  cosines, 

8*wmWl     b*  +  c2-a2\     bcf1      fr2  +  c2  -  a2^ 
2\  26c       y      2^  26c       J 

^2&c  +  &2  +  c2-a2     2  5c  -  b2  -  c2  +  a2 

4  '  4 

_6+_c_-j-a     5-f-c  — a     a—b  +  c  (  a+J^c> 
~        2         *        2         '        2         *  "      2 

which  may  be  written  in  the  form 

S2  =  s(s-a)(s-b)(s-c), 

where  2s  =  a  +  6  +  c.     Therefore, 


(2)  S  =  Vs(s  -a)(s-  b)  (s-c). 

f  35^  The  Radius  of  the  Inscribed  Circle.  If  r  is  the  radius 
of-£ne  inscribed  circle,  we  have  from  elementary  geometry, 
since  s  is  half  the  perimeter  of  the  triangle,  S  =  rs ;  equating 
this  value  of  8  to  that  found  in  equation  (2)  of  the  last  article 
and  then  solving  for  r,  we  get, 


-v 


(s  —  a)(s  —  b)(s  —  c) 
s 


IV,  §  36] 


OBLIQUE  TRIANGLES 


47 


EXERCISES 

Find  the  area  of  the  triangle  ABC,  given 
\>  1,   a  =  25,  b  =  31.4,  C  =  80°  25'.  4.   a  =  10,  b  =  7,     C  =  60°. 

2-    &  =  24,   c  =  34.3,  J.  =  60°  25'.        N»  5.    a  =  10,  b  =  12,   C  =  60°. 
3.    a  =  37,  6  =  13,       C  =  40°.  ^  6.    a  =  10,  6  =  12,   C  =  8\ 

7.  Find  the  area  of  a  parallelogram  in  terms  of  two  adjacent  sides 
and  the  included  angle. 

8.  The  base  of  an  isosceles  triangle  is  20  ft.  and  the  area  is  100/V3 
sq.  ft.  Find  the  angles  of  the  triangle.  Ans.  30°,  30°,  120°. 
\j  9.  Find  the  radius  of  the  inscribed  circle  of  the  triangle  whose  sides 
are  12,  10,  8. 

10.  How  many  acres  are  there  in  a  triangular  field  having  one  of  its 
sides  50  rods  in  length  and  the  two  adjacent  angles,  respectively,  70° 
and  60°  ? 


and  60°  \ 
3,1 


next 


The  Law  of  Tangents. 

chapter  the  formulas  in  this 
and  the  next  article  will  be 
needed. 

Let  CD  be  the  bisector  of 
the  angle  G  of  the  A  ABC. 
Through  A  draw  a  line  II  DC, 
meeting  BC  produced  in  E. 
Then  CE  =  b.  Why  ?  From 
A  draw  a  line  q  X  DC  meeting 

CB  in  F.     At  F  draw  a  line  r  J_  AF  meeting  AB  in  G. 
AE=p. 

Now  AACF  is  isosceles.  Why?  The  angle  ACE  =  ZA 
+  /.B  and  the  bisector  of  Z.ACE  is  _L  CD.  Hence  Z  CAF 
=  Z  CFA  =  ±Z(A  +  B).  Moreover  Z BAF=  ZA-±Z(A 
+  B)  =  ±Z(A-B). 


Let 


Now 


tan 


A  +  B 


and  tan 


tan 


tan 


A  +  B 


A-B 


48  PLANE  TRIGONOMETRY  [IV,  §  36 

But  £  =  !f  =  «  +  *.     Why? 

tan 
Hence 


tan 


a, 


.   Angles  of  a  Triangle  in  Terms  of  the  Sides.     Con- 


f  struct  the  inscribed  circle  of  the  triangle 

and.  denote  its  radius  by  r..     If  the  perim- 
eter a  +  6  +  c  =  2s,  then  (Fig.  36) 

AE  =  AF=s  -a. 
BD  =  BF=s-b. 
CD=CE  =  s-c.     . 
_  i  ti        r         .  ■  ,  >-        r 


Then     tan  i  ^4  = ,      tan  \  B  — ,      tan  I  C  = 

s  —  a  j     s  —  b 


where,  from  §  35,  rrUr^L        /\   . 

=  J(s-aXs-b)(s-c)_, 


A  F  tc  £+£>/*  VA  A  ->S-^i>  +  <WJ>  M 


38.   Solution   of  Triangles  by  Means  of  the  Haversine. 

The  haversine  may  be  used  advantageously  in  the  solution  of  triangles, 
(1)  when  two  sides  and  the  included  angle  are  given  ;  (2)  when  the 
three  sides  are  given.    The  law  of  cosines  gives 

2havJ.  =  1  -  cos^l  =  1  -  &2  +  c2-a2 

2  6c 

_q2_(fr  _  c)2 

2  be 
or  4  6chav  A  =  a2  —  (b  —  c)2. 

1.  If  6,  c  and  A  are  given  we  may  find  a  from  the  formula 

(1)  a2=(b-c)2  +  Ibch&vA. 

Similar  formulas  give  b2  or  c2  in  terms  of  a,  c,  .B  and  a,  6,  C  respectively. 

2.  If  a,  6,  c  are  given,  we  may  find  A  from  the  formula 

(2)  hav  A  =  *,-<»- «)'  =  .('-W-Q  • 
w  4  be  6c 

Similar  formulas  will  give  B  and  0, 


IV,  §  38]  OBLIQUE  TRIANGLES  49 


Example  1.     Given  A  =  94° 

23'.4,  b  =  55.12,  c  =  39.90.     To  find  . 

By  formula  (1)  above  : 

6  =      55.12 

be  =  2199 

c  =      39.90 

hav  94°  23'.4  =  0.0446 

(6-c)  =      15.22 

be  hav  A=  1184 

(6-c)2  =    231.6 

4  6c  hav  J.  =  4736 

4  be  hav  A  =  4736 

a*  =  4968 

» 

a  =  70.49 

Example  2.     Given  a  =  4.51 

,  6  =  6.13,  c  =  8.16.     FindJL,  B,  C. 

a2=    2034         hav  ^  =  1«^  =  0.0811        A=    33*05 
(6-c)2=      4.12                         200.1 

a2_(6-c)2=    16.22 

be  =    50.02 

4  6c  =  200.1 

62=    37.58        hav  5  =  2426  =0.1648       B  =    47°  54 
(C_a)2=    13.32                        147.21 

62  _  (C  _  a)2  _    24.26 

ac=    36.80 

4ac  =  147.21 

C2=    66.59         hav  C  -  63,97  -  0.5785       C --    99°  02' 
(6-a)2=      2.62                        110.58             Check;     18QO  0, 
C2«(6_a)2=    63.97 

ab=    27.646 

4  a&  =  110.58 

EXERCISES 

Solve  the  following  triangles  : 

^     1.     a  =  62.1,  6  =  32.7,     c  =  47.2. 

^  2.   vl  =  37°20',       6  =  2.4,       c  =  4.7. 

N    3.    B  =  121°  32',    a  =  27.9,     c  =  35.8. 

^    4.     a  =  3.2,  6  =  5.7,       c  =  6.5. 

5.  C  =  72°21'.4,  a  =  314.1,  6  =  427.3. 

6.  a  =  346.1,         6  =  425.8,   c  =  562.3. 


CHAPTER  V 


39.  The  Invention  of  Logarithms.  The  extensive  numeri- 
cal computations  required  in  business,  in  science,  and  in  engi- 
neering were  greatly  simplified  by  the  invention  of  logarithms 
by  John  Napier,  Baron  of  Merchiston  (1550-1617).  By  means 
of  logarithms  we  are  able  to  replace  multiplication  and  division 
by  addition  and  subtraction,  processes  which  we  all  realize  are 
more  expeditious  than  the  first  two. 

If  we  consider  the  successive  integral  powers  of  2 


a) 


Exponent  x 

1 

2 

3 

4 

5 

6 

7 

Result  2*     .     . 

2 

4 

8 

16 

32 

64 

128 

Exponent  x 

8 

9 

10 

11 

12 

etc. 

A.  P. 

Result  2*     .     . 

256 

512 

1024 

2048 

4096 

etc. 

G.  P. 

we  see  that  the  results  form  a  geometric  progression  (G.  P.) 
and  the  exponents  an  arithmetic  progression  (A.  P.).  We 
know  from  elementary  algebra  that 


and 


xn 
xn 


Hence  if  we  wish  to  multiply  two  numbers  in  our  G.  P.  e.g. 
4  x  8,  we  merely  have  to  add  the  corresponding  exponents  2. 
and  3  and  under  the  sum  5  find  the  desired  product  32.  Sim- 
ilarly, if  we  wish  to  divide  e.g.  4096  by  128,  we  merely  have  to 
subtract  the  exponent  corresponding  to  128,  from  that  cor- 

50 


V,  §  39] 


LOGARITHMS 


51 


responding  to  4096  and  under  their  difference  5  we  find  the 
desired  quotient  32. 

To  make  the  above  plan  at  all  useful  it  is  evident  that  our 
table  must  be  expanded  so  as  to  contain  more  numbers.  First 
we  can  expand  our  table  so  that  it  will  contain  numbers  less 
than  2,  by  subtracting  1  successively  from  the  numbers  in  the 
A.  P.  and  by  dividing  successively  by  2  the  numbers  in  the 
G.P. 


(2) 


In  the  second  place  we  may  find  new  numbers  by  inserting 
arithmetic  means  and  geometric  means.  Thus,  if  we  take  the 
following  portion  of  the  preceding  table 


-5 

-4 

-3 

-2 

-1 

0 

1 

2 

3 

4 

5 

6 

7 

0.03125 

0.0625 

0.125 

0.25 

0.5 

1 

2 

4 

8 

16 

32 

04 

128 

-2 

-  1 

0 

1 

2 

3 

4 

* 

| 

1 

2 

4 

8 

16 

and  insert  between  every  two  successive  numbers  of  the  upper 
line  their  arithmetic,  and  between  every  two  successive  num- 
bers of  the  lower  line  their  geometric  mean,  we  obtain  the 
table 


(3) 

-2 

-f 

-1 

-i 

0 

± 

1 

1 

2 

5 

3 

i 

4 

i 

}V2 

1 

^V2 

1 

V2 

2 

2V2 

4 

4V2 

8 

8V2 

16 

If  the  radicals  are  expressed  approximately  as  decimals,  this 
table  takes  the  form 

-2.C 

-1. 

5-l.< 

)-0.5 

0 

0.5 

1.0 

1.5 

2 

2.5 

3 

3.5 

4 

0.25 

0.35 

0.50 

0.72 

1.00 

1.41 

2.00 

2.83 

4.00 

5.66 

8.00 

11.31 

16 

52  PLANE  TRIGONOMETRY  [V,  §  39 

By  continuing  this  process  we  can  make  any  number  appear 
in  the  G.  P.  to  as  high  a  degree  of  approximation  as  we  desire. 
To  prepare  an  extensive  table,  which  gives  values  at  small  inter- 
vals, is  quite  laborious.  However,  it  has  been  done,  and  we 
have  printed  tables  so  complete  that  actual  multiplication  of 
any  two  numbers  can"  be  replaced  by  addition  of  two  other 
numbers.     We  shall  soon  learn  how  to  use  such  tables. 

40.   Definition    of  the   Logarithm.    The   logarithm  of   a 

number   JV  to   a   base   b  (b  >  0,    =£  1)   is   the  exponent  x   of 

the  power  to  which  the  base  b  must  be  raised  to  produce  the 

number  JV. 

That  is,  if 

&*=  N, 
then 

x  ^lo&AT. 

These  two  equations  are  of  the  highest  importance  in  all  work 
concerning  logarithms.  One  should  keep  in  mind  the  fact 
that  if  either  of  them  is  given,  the  other  may  always  be 
inferred. 

The  numbers  forming  the  A.  P.  in  tables  1,  2,  and  3  of  §  39 
are  the  logarithms  of  the  corresponding  numbers  in  the  G.  P., 
the  base  being  2.  From  table  3  we  have  2*  =  4  V2  which  says 
log24V2  =  |. 

EXERCISES 

1.  When  3  is  the  base  what  are  the  logarithms  of  9,  27,  3,  1,  81,  |, 

2.  Why  cannot  1  be  used  as  the  base  of  a  system  of  logarithms  ? 

3.  When  10  is  the  base  what  are  the  logarithms  of  1,  10,  100,  1000  ? 

4.  Find  the  values  of  x  which  will  satisfy  each  of  the  following 
equalities : 

(a)  log3  27  =  x.  (d)  loga  a  =  x.  (g)  log2  x  =  6. 

(6)  \ogx  3  =  1.  (e)  loga  l=x.  (h)  log32z  =  |. 

(c)  log,  5=|.  (/)  log,  b\  =  x.  ('J)  logo.001  x  =  2. 


V,  §  41]  LOGARITHMS  53 

5.    Find  the  value  of  each  of  the  following  expressions  : 

(a)  log  2 16.  (c)  loge^  (e)  log25 125. 

(6)  log34349.  (d)  log2Vl6.  (/)  log2lfr. 

41.  The  Three  Fundamental  Laws  of  Logarithms.  From 
the  laws  of  exponents  we  derive  the  following  fundamental 
laws. 

I.  TJie  logarithm  of  a  product  equals  the  sum  of  the  logarithms 
of  its  factors.     Symbolically, 

log6  MN  =  log6  M  +  log6  N. 

Proof.  Let  log6  M  =  x,  then  bx  =  M.  Let  log6  N=  y,  then 
6V  =  N.     Hence  we  have  MN  =  bx+y,  or 

log6  MN  ax  m  +  y,   i.e.   log6  MN  =  log,,  M  +  log6  N. 

II.  Tlie  logarithm  of  a  quotient  equals  the  logarithm  of  the 
dividend  minus  the  logarithm  of  the  divisor.     Symbolically, 

log6^f=  log6  M  -  log&  N. 

N 

Proof.  Let  log6  M  =  x,  then  b*  =  M.  Let  log6  N—yy  then 
b'J  m  N.     Hence  we  have  M/N=  b*'",  or 

M  M 

^ogb-  =  x-y,    i.e.    \ogb^.  =  \ogb  M  -  \ogb  N. 

III.  The  logarithm  of  the  pth  power  of  a  number  equals  p 
times  the  logarithm  of  the  number.     Symbolically 

logfe  Mp  =  p  log6  M. 

Proof.  Let  log6  M  =  x,  then  bx  =  M.  Raising  both  sides 
to  the  pth  power,  we  have  bpx  =  M v.     Therefore 

log6  M p  =px=p  log,  M. 

Prom  law  III  it  follows  that  the  logarithm  of  the  real  positive 
nth  root  of  a  number  is  one  nth  of  the  logarithm  of  the  number. 


54  PLANE  TRIGONOMETRY  [V,  §  41 

2  EXERCISES 

Given  logi02  =  0.3010,   log10  3  =  0.4771,  logio  7  =  0.8451,  find  the 
of  each  of  the  f  ollowingg|rxpressions  : 
(a)   logw6.  (/)  logi05. 

[Hint:  logio  2x3=  log10  2  +  logio  3.]      [Hint:  log10  5  =  log10  y.J 

(6)  logio  21.0.  (?)  logio  m 

(c)  logio  20.0.  (h)  logio  Vl4. 

(d)  logio  0.03!  (i)   logio  49^_ 

(e)  logio  |.  (i)  logio  V24.7&. 

2.    Given  the  same  three  logarithms  as  in  Ex.  1,  find  the  value  of  each 
of  the  following  expressions  :  r>  * 

/„\    u„    4  x  5  x  7  ,,x  •  '     5  x  3  x  20  f  *    ,        2058 

(a)  IogI°^2T^-         (6)  logl°-^T^-       N(c)  loSl0^i- 

^(d)   logio  (2)*.  (e)  logic  (3)8(5)«,  (/)  logio(23)Q). 


<5> 


Logarithms  to  the  Base  10.  Logarithms  to  the  base  10 
are  known  as  common  or  Briggian  logarithms.  Proceeding  as 
in  §  39  we  can  show  that  100-3010  =  2,  i.e.  log10  2  =  0.3010.  Let 
ns  multiply  both  members  of  the  equation  1003010  ==  2  by  10,  102, 
103,  etc.  and  notice  the  effect  on  the  logarithm. 
10o.3oio  =  2  log10  2  =  0.3010 

10  3010  =  20  log10  20  =  1.3010 

102.3oio  =  200  logL0  200  =  2.3010. 

It  should  be  clear  from  this  example  that  the  decimal  part  of 
the  logarithm  (called  the  mantissa)  of  a  number  greater  than  1 
depends  only  on  the  succession  of  figures  composing  the  num- 
ber and  not  on  the  position  of  the  decimal  point,  ^vhile  the  in- 
tegral part  (called  the  characteristic)  depends  simply  on  the 
position  of  the  decimal  point.  Hence  it  is  only  necessary  to 
tabulate  the  mantissas,  for  the  characteristics  can  be  found  by 
inspection  as  the  following  considerations  show. 

Since 
10°  =  1,    lO^lO,    102  =  100,    103  =  1000,    104  =  10,000,  etc. 
we  have      logj0 1  =  0,  log10 10  =  1,  log10 100  =  2, 

log,o  1000  =  3,  logM  10,000  =  4,  etc. 


V,  §  42]  LOGARITHMS  55 

It  follows  that  a  number  with  one  digit  (=f=  0)  at  the'  left  of  the 
decimal  point  has  for  its  logarithm  a  number  equal  to  0  4-  a 
decimal ;  a  number  with  two  digits  at  the  left  of  its  decimal 
point  has  for  its  logarithm  a  number  equal  to  1  -+-  a  decimal ;  a 
number  with  three  digits  at  the  left  of  the  decimal  point  has 
for  its  logarithm  a  number  equal  to  2  +  a  decimal,  etc.  We 
conclude,  therefore,  that  the  characteristic  of  the  common  loga- 
rithm of  a  number  greater  than  1  is  one  less  than  the  number  of 
digits  at  the  left  of  the  decimal  point. 

Thus,  logio  456.07  =  2.65903. 

The  case  of  a  logarithm  of  a  number  less  than  1  requires 
special  consideration.  Taking  the  numerical  example  first  con- 
sidered above,  if  log10  2 =0.30103,  we  have  log10  0.2=0.30103-1. 
Why?  This  is  a  negative  number,  as  it  should  be  (since  the 
logarithms  of  numbers  less  than  1  are  all  negative,  if  the 
base  is  greater  than  1).  But,  if  we  were  to  carry  out  this 
subtraction  and  write  log10  0.2  =  —  0.69897  (which  would  be 
correct),  it  would  change  the  mantissa,  which  is  inconvenient. 
Hence  it  is  customary  to  write  such  a  logarithm  in  the  form 
9.30103  -  10. 

If  there  are  n  ciphers  immediately  following  the  decimal 
point  in  a  number  less  than  1,  the  characteristic  is  —  n—  1. 
For  convenience,  ifn<  10,  we  write  this  as  (9  —  n)  —  10.  TJiis 
characteristic  is  written  in  two  parts.  The  first  part  9  —  n  is 
ivritten  at  the  left  of  the  ma?itissa  and  the  —  10  at  the  right. 

In  the  sequel,  unless  the  contrary  is  specifically  stated,  we 
shall  assume  that  all  logarithms  are  to  the  base  10.  We  may 
accordingly  omit  writing  the  base  in  the  symbol  log  when  there 
is  no  danger  of  confusion.  Thus,  the  equation  log  2  =  0.30103 
means  log10  2  =  0.30103. 

To  make  practical  use  of  logarithms  in  computation  it  is 
necessary  to  have  a  conveniently  arranged  table  from  which 
we  can  find  (a)  the  logarithm  of  a  given  number  and  (b)  the 
number   corresponding   to  a   given   logarithm.     The   general 


./  <* 


) 


56  PLANE  TRIGONOMETRY  [V,  §  42 

principles  governing  the  use  of  tables  will  be  explained  by  the 
following  examples  [Tables,  pp.  110,  111]. 

Example  1.     Find  log  42.7. 

The  characteristic  is  1.  In  the  column  headed  N  (p.  110)  we  find  42 
and  if  we  follow  this  row  across  to  the  column  headed  7,  we  read  6304, 
which  is  the  desired  mantissa.     Hence  log  42.7  =  1.6304. 

Example  2.     Find  log  0.03273. 

The  characteristic  is  8  —  10.  The  mantissa  cannot  be  found  in  our 
table,  but  we  can  obtain  it  by  a  process  called  interpolation.  We  shall 
assume  that  to  a  small  change  in  the  number  there  corresponds  a  propor- 
tional change  in  the  mantissa.     Schematically  we  have 


u     '  ^-'    ,     Number  Mantissa 


difference  =  10 


.  T3270        ->        5145" 
L3273        ->  ?         4  =  difference 


3280        — >»         5159  J 


Our  desired  mantissa  is  5145  +  ^-14  =  5149.  Hence  log  0.03273 
=  8.5149  -  10. 

Example  3.     Find  x  when  log  x  —  0.8485. 

We  cannot  find  this  mantissa  in  our  table,  but  we  can  find  8482  and 
8488  which  correspond  to  7050  and  7060  respectively.  Reversing  the 
process  of  example  2,  we  have  schematically 

Number  Mantissa 

"7050        <-  84821  _" 

Difference  =  10       ?  <—  8485  J         6  =  difference 

7060        <-  8488 

Hence  the  significant  figures  in  our  required  number  are  7050  -f- 1  •  10 
=  7055.     Since  the  characteristic  is  0  the  required  number  is  7.055. 


EXERCISES 

/^)  Find  the  logarithms  of  the  following  numbers  from  the  table  on 
ppYllO,  111  :  482,  26.4,  6.857,  9001,  0.5932,  0.08628,  0.00038. 

2.   Find  the   numbers    corresponding   to   the    following   logarithms : 
2.W35,  0.3502,  7.9599  -  10,  9.5300  -  10,  3.6598,  1.0958. 

43.  Use   of  Logarithms   in    Computation.     The   way   in 

which  logarithms  may  be  used  in  computation  will  be  suffi- 
ciently explained  in  the  following  examples.  A  few  devices 
often  necessary  or  at  least  desirable  will  be  introduced.     The 


V,  §  43]  LOGARITHMS  57 

latter  are  usually  self-explanatory.  Reference  is  made  to 
them  here,  in  order  that  one  may  be  sure  to  note  them  when 
they  arise.  The  use  of  logarithms  in  computation  depends,  of 
course,  on  the  fundamental  properties  derived  in  §  41. 

Example  1.     Find  the  value  of  73.26  x  8.914  x  0.9214. 

We  find  the  logarithms  of  the  factors,  add  them,  and  then  find  the 
number  corresponding  to  this  logarithm.  The  work  may  be  arranged  as 
follows : 


Numbers 

Logarithms 

73.26 

(-►) 

1.8649 

8.914 

(-» 

0.9501 

0.9214 

(-» 

9.9645  -  10 
12.7795  -  10 

Product  =  601.9  Arts. 

(«-) 

2.7795 

Example  2.     Find  the  value  of  732.6  • 

4-  89.14. 

Numbers 

Logarithms 

732.6 

(-*) 

2.8649 

89.14 

(-*0 

1.9501 

Quotient  =  8.219  Ans. 

(«-) 

0.9148 

Example  3.     Find  the  value  of  89.14 

-=-  732.6. 

Numbers 

Logarithms 

89.14 

c-*o 

11.9501  -  10 

732.6 

(->) 

2.8649 

Quotient  =  0.1217  Ans. 

«-) 

9.0852  -  10 

Example  4.     Find  the  value  of          * 

x  21.63 

Whenever  an  example  involves  several  different  operations  on  the 
logarithms  as  in  this  case,  it  is  desirable  to  make  out  a  blank  form.  When 
a  blank  form  is  used,  all  logarithms  should  be  looked  up  first  and  entered 
in  their  proper  places.  After  this  has  been  done,  the  necessary  opera- 
tions (addition,  subtraction,  etc.)  are  performed.  Such  a  procedure 
saves  time  and  minimizes  the  chance  of  error. 

Form 

Numbers  Logarithms 

763.3  (-►)              

21.63  (-»     (  +  ) 

product  

986.7  (-»  (-)..... 

....  Ans.       (<-)              


58 


PLANE  TRIGONOMETRY 


[V,  §43 


Form  Filled  In 

Numbers 

Logarithms 

763.2 

(-» 

2.8826 

21.63 

(-») 

1.3351 

product 

4.2177 

986.7 

(-*) 

2.9942 

16.73    Ans. 

«-) 

1.2235 

Example  5.     Find  (1.357)5. 

Numbers 

Logarithms 

1.357 

(-» 

0.1326 

(1.357)5  =  4.602 

Ans. 

(-*-) 

0.6630 

Example  6.     Find  the  cube 

root  of  30.11. 

Numbers 

Logarithms 

30.11 

(-» 

1.4787 

#30.11  =3.111 

Ans. 

(«-) 

0.4929 

Example  7.     Find  the  cube  root  of  0.08244. 

Numbers  Logarithms 

0.08244  (->)  28.9161  -  30 

#0.08244  =  0.4352    Ans.   («<-)     9.6387-10 


6   D 


EXERCISES 


Compute  the  value  of  each  of  the  following  expressions  using  the  table 
on  pp.  110,  111. 


1.  34.96  x  4.65. 

2.  518.7  x  9.02  x  .0472. 

3  0.5683 _ 
0.3216* 

4  5.007  x  2.483 
6.524  x  1.110* 

5.  (34.16  x  .238)2. 

6.  8.572  x  1.973  x  (.8723)2. 


K.# 


8076  x  3.184 


(2.012)5 
O  10.   a/2941  ><  17- 


11. 


'2173  x  18.75 
#0.00732 
#735 
^12.    (20.027)* 
d/lS.    21(». 


■'  i 


e 


648.8 


'(21.4)1 

/1379 
>2791 " 
/ 


v14.    Vio^.ioo2. 
15.    (0.02735)*. 

die. 


f  A 


#3275 


(2.01) 


y 


i 


V,  §  44]  LOGARITHMS  59 

44.  Cologarithms.  Since  —  and  M  •  — -  are  equivalent, 
we  may  in  a  logarithmic  computation,  add  the  logarithm  of 
—    instead   of   subtracting  log  N.     The   logarithm   of  -—   is 

called  the  cologarithm  of  N.     Therefore 

colog  N  =  log  1/N  =  log  1  —  log  N  =  —  log  N, 

since  log  1  is  zero. 

We  write  cologarithms,  like  logarithms,  with  positive  man- 
tissas. Therefore  the  cologarithm  is  most  easily  found  by  sub- 
tracting the  logarithm  from  zero,  written  in  the  form  10.0000 
-10. 

Example.     Find  the  colog  27.3. 

10.0000  -  10 

i  log  27. 3=    1.4362 


colog  27.3=    8.5638-10 

The  cologarithm  can  be  written  down  immediately  by  subtracting  the 
last  significant  figure  of  the  logarithm  from  10  and  each  of  the  others 
from  9.  If  the  logarithm  is  positive  the  cologarithm  is  negative  and 
hence  —  10  is  affixed. 

There  is  no  gain  in  using  cologarithms  when  we  have  a  quotient  of  two 
numbers.  There  is  an  advantage  when  either  the  numerator  or  denomi- 
nator contains  two  or  more  factors,  for  we  can  save  an  operation  of  addi 
tion  or  subtraction.     Let  us  solve  Ex.  4,  §  43,  using  cologarithms. 

Example.     Find  the  value  of  763'2  x  2L63 ■ 

986.7 

Numbers  Log 

763.2  .->■  2.8826 

21.63  ->  1.3351 

986.7  ->  (colog)  7.0058  - 10 

16.73  <-  1.2235 

EXERCISES 

Compute  the  value  of  each  of  the  following  expressions,  using  cologa- 
rithms. 


/?\  J2.80  x  37.6  /"T\  J 


97.63  x  876.5 
2876  x3.4  x  2.987 


60  PLANE  TRIGONOMETRY  [V,  §44 

3  5  5  V3275        , 

'    7  x  8  x  9  x  27.6  ^J  (2.01)*(1.76)» 

4.       312    •  6        1293  x  12  7  x  5 

610,27  N^(l  +  2V3)(760  +  8)' 

MISCELLANEOUS  EXERCISES 

1.  What  objections  are  there  to  the  use  of  a  negative  number  as  the 
base  of  a  system  of  logarithms  ? 

2.  Show  that  al°sax  =  x. 

3 .  Write  each  of  the  following  expressions  as  a  single  term  : 
'a)   log x  +  log y  —  log  z.  QpS^P  log x  —  2  log y  +  3  log  z. 


XcpS  log  a  —  log  (x  +  y)  -  \  log  (ex  +  tf)'+  log  Vw  +  x. 

4}    Solve  for  x  the  following  equations  : 

§2  log2  £  +  log2  4  =  1.  (c)  2  logio  x  -  3  log10  2  =  4. 

log3  x  -  3  log3  2  =  4.  (d)  3  log2  x  +  2  log2  3  =  1,    - 

/5.   How  many  digits  are  there  in  235  ?  3142  ?  312  x  2»  ?  ^g| 

y6.    Which  is  the  greater,  (f£)100  or  100  ? 
/  7>  Find  the  value  of  each  of  the  following  expressions : 
(a%  log6  35.  ((py  log3  34.  (g)  log7  245.  (d)  log13  26.     - 

8.  Prove  that  logb  a  •  loga  6  =  1. 

9.  Prove  that 


log„a;  +  Vx2~-  =  2  logo[x  +  Vx2  -  1]. 
«  —  Vx2  —  1 

10.    The  velocity  v  in  feet  per  second  of  a  body  that  has  fallen  s  feet 
is  given  by  the  formula  v  =  V64.3s. 

What  is  the  velocity  acquired  by  the  body  if  it  falls  45  ft.  7  in.  ? 
/ 11.   Solve  for  x  and  ?/  the  equations  ;  2X  =  16v,  x  +  4  ?/  =  4. 


► 


m 


CHAPTER  VI 
LOGARITHMIC  COMPUTATION 

46.  Logarithmic  Computation.  In  the  last  chapter  a  few 
examples  of  the  use  of  logarithms  in  computation  were  given 
in  connection  with  a  four-place  table.  Such  a  table  suffices 
for  data  and  results  accurate  to  four  significant  figures.  When 
greater  accuracy  is  desired  we  use  a  five-,'  six-,  or  seven-place 
table. 

No  subject  is  better  adapted  to  illustrate  the  use  of  logarith- 
mic computation  than  the  solution  of  triangles,  which  we  shall 
consider  in  some  detail.  Five-place  tables  and  logarithmic 
solutions  ordinarily  are  used  at  the  same  time,  since  both  tend 
toward  greater  speed  and  accuracy. 

46.  Five-place  Tables  of  Logarithms  and  Trigonometric 
Functions.  The  use  of  a  five-place  table  of  logarithms  differs 
from  that  of  a  four-place  table  in  the  general  use  of  so-called 
"  interpolation  tables "  or  "  tables  of  proportional  parts,"  to 
facilitate  interpolation.  Since  the  use  of  such  tables  of  pro- 
portional parts  is  fully  explained  in  every  good  set  of  tables, 
it  is  unnecessary  to  give  such  an  explanation  here.  It  will  be 
assumed  that  the  student  has  made  himself  familiar  with  their 
use.* 

In"  the  logarithmic  solution  of  a  triangle  we  nearly  always 
need  to  find  the  logarithms  of  certain  trigonometric  functions. 
For  example,  if  the  angles  A  and  B  and  the  side  a  are  given, 
we  find  the  side  b  from  the  law  of  sines  given  in  §  30, 


,  _  a  sin  B 
♦^*  sin  A 

*  For  this  chapter,  such  a  five-place  tahle  should  be  purchased.  See,  for 
example,  The  Macmillan  Tables,  which  contain  all  the  tables  mentioned 
here  with  an  explanation  of  their  use. 

61 


62  PLANE  TRIGONOMETRY  [VI,  §  46 

To  use  logarithms  we  should  then  have  to  find  log  a,  log  (sin  B) 
and  log  (sin  A).  With  only  a  table  of  natural  functions  and  a 
table  of  logarithms  at  our  disposal,  we  should  have  to  find  first 
sin  A,  and  then  log  sin  A.  For  example,  if  A  =  36°  20',  we 
would  find  sin  36°  20'  =  0.59248,  and  from  this  would  find  log 
sin  36°  20'  =  log  0.59248  m  9.77268  -  10.  This  double  use  of 
tables  has  been  made  unnecessary  by  the  direct  tabulation  of  the 
logarithms  of  the  trigonometric  functions  in  terms  of  the  angles. 
Such  tables  are  called  tables  of  logarithmic  sines,  logarithmic 
cosines,  etc.  Their  use  is  explained  in  any  good  set  of  tables. 
The  following  exercises  are  for  the  purpose  of  familiarizing 
the  student  with  the  use  of  such  tables. 

J  EXERCISES 

V.   Find  the  following  logarithms  :  * 

(a)  log  cos  27°  40'.5.  (d)  log  ctn  86°  53'. 6. 

(6)  log  tan  85°  20'.2.  (e)  log  cos  87°  6'.2. 

\}c)  log  sin  45°  40'. 7.  (/)  log  cos  36°  53'. 3. 
"■■k.   Find  A,  when 

(a)  log  sin  A  =  9.81632  -  10.  (d)  log  sin  A  =  9.78332  -  10.     • 

(6)  log  cos  A  =  9.97970  -  10.  (e)  log  ctn  }  A  =  0.70352. 

(c)  log  tan  A  =  0.45704.  •(/)  log  tan  \A  =  9.94365  -  10. 

VL   Find  Mf  tan  fl  =  476-32x89-710. 
\  87325 

^  4.   Given  a  triangle  ABC,  in  which  ZA  =  32°,  Z  B  =  27°,  a  =  5.2,  find 

b  by  use  of  logarithms. 

47.  The  Logarithmic  Solution  of  Triangles.  The  effective 
use  of  logarithms  in  numerical  computation  depends  largely  on 
a  proper  arrangement  of  the  work.  In  order  to  secure  this, 
the  arrangement  should  be  carefully  planned  beforehand  by 
constructing  a  blank  form,  which  is  afterwards  filled  in.  More- 
over, a  practical  computation  is  not  complete  until  its  accuracy 
has  been  checked.  The  blank  form  should  provide  also  for  a 
good  check.     Most  computers  find  it  advantageous  to  arrange 

*  Five-place  logarithms  are  properly  used  when  angles  are  measured  to  the 
nearest  tenth  of  a  minute.  For  accuracy  to  the  nearest  second,  six  places 
should  be  used. 


VI,  §  48]         LOGARITHMIC  COMPUTATION  63 

the  work  in  two  columns,  the  one  at  the  left  containing  the 
given  numbers  and  the  computed  results,  the  one  on  the  right 
containing  the  logarithms  of  the  numbers  each  in  the  same 
horizontal  line  with  its  number.  The  work  should  be  so 
arranged  that  every  number  or  logarithm  that  appears  is 
properly  labeled  ;  for  it  often  happens  that  the  same  number 
or  logarithm  is  used  several  times  in  the  same  computation  and 
it  should  be  possible  to  locate  it  at  a  glance  when  it  is  wanted. 
The  solution  of  triangles  may  be  conveniently  classified 
under  four  cases : 

Case  I.     Given  two  angles  and  one  side. 

Case  II.  Given  two  sides  and  the  angle  opposite  one  of  the 
sides. 

Case  III.     Given  two  sides  and  the  included  angle. 

Case  IV.     Given  the  three  sides. 

In  each  case  it  is  desirable  (1)  to  draw  a  figure  representing 
the  triangle  to  be  solved  with  sufficient  accuracy  to  serve  as  a 
rough  check  on  the  results  ;  (2)  to  write  out  all  the  formulas 
needed  for  the  solution  and  the  check ;  (3)  to  prepare  a  blank 
form  for  the  logarithmic  solution  on  the  basis  of  these 
formulas ;  (4)  to  fill  in  the  blank  form  and  thus  to  complete 
the  solution. 

We  give  a  sample  of  a  blank  form  under  Case  I ;  the  student 
should  prepare  his  own  forms  for  the  other  cases. 

48.   Case  I.     Given  Two  Angles  and  One  Side. 

Example.     Given:  a=430.17,  ^1=47°  13'.2,  B=52°  29'.5.     (Fig.  37.) 
To  find:  C,  6,  c. 
Formulas : 

C  =  180°-(A  +  B), 

b=—2-sinB, 
sin  A 

sin  C. 


sin  A 

Check  (§  36):  ^=±  =  tan  $(<?-*)  . 

^      J    c  +  b      tanJ(C+B)  *  Fig.  37 


64 


PLANE  TRIGONOMETRY 


[VI,  §  48 


The  following  is  a  convenient  blank  form  for  the  logarithmic  solu- 
tion. The  sign  (+)  indicates  that  the  numbers  should  be  added  ;  the 
sign  (— )  indicates  that  the  number  should  be  subtracted  from  the  one 
just  above  it. 


A  = 

(  +  )*  = 
A+  B  = 

C  = 

a  = 
sin  A  = 


Numbers 


179°  60'.0 


Logarithms 


sin 


a/sin  A 
sin  B  =  sin 
b  =  .     . 


a/sin  A 

sin  C 

c 


c-b  = 
c+  b  = 


C-B=.     . 

C+ B= .  . 
tan  |  (  C  —  B)  =  tan 
tan  \{C  +  B)=  tan 


(-) 


-»)   (  +  ) 

-H  (+) 

Check 

■»   (-) 


•)    (-) 


(1) 

(Logs  (1)  and  (2) 
.  should  be  equal 
.       for  check.) 
"(2) 


Filling  in  this  blank  form,  we  obtain  the  solution  as  follows. 


Numbers 

A=    47°13'.2 

B=    52°29'.6 

A+  B=    99°42'.8 

179°  60'.0 


Logarithms 


0=    80°17'.2 

a^=  430.17 
sin  A  =sin47°13'.2 
a/sin  A 
sin  B  =  sin  52°  29'.  6 
b  =  464.94  Ans. 


2.63364 
(-)  9.86567  -  10 

2.76797 
(  +  )  9.89943  -  10 

2.66740 


Check* 


VI,  §  49]  LOGARITHMIC  COMPUTATION  65 

a/sin  A  2.76797 

sin  C  =  sin 80°  17'. 2     (->)     (  +  )  9.99373-  10 
c  =  577.70  Ans.     (<-)  2.76170 

Check 
c-b  =    112.76  (->)  2.05215 

c  +  b  =  1042.64  (->)     (-)  3.01813 

9.03402  -  10 
C-B  =    27°47'.6 
C+£  =  132°46'.8 
tan|(C-  i*)=tanl3°53'.8    (->)  9.39342-10 

tan£(C  +  5)=  tan 66° 23'. 4    (->-)     (-)  0.35942 

9.03400  -  10 

EXERCISES 
Solve  *ud  ulUWft  the  following  triangles  ABC : 
.  V.    a  =  372.5,  ^4  =  25°  30',  5  =  47°  50'. 

>*  X  c  =  327.85,  A  =  110° 52'.9,  5  =  40° 31'.7.        Ans.     C  =  28° 35'.4, 

a  =  640.11,  6  =  445.20. 
3.  a  =  53.276,  A  =  108°  50'.0,  C  =  57°  13'. 2. 
^  V  b  =  22.766,  B  =  141°  59M,  C  =  25°  12'.4. 
5.  b  =  1000.0,  B  =  30°  30'.5,  C  =  50°  50'.8. 
X,  «'  a  =  257.7,  J.  =  47°  25',  B  =  32°  26'. 

49.  Case  n.  Given  Two  Sides  and  an  Angle  Opposite 
One  of  Them. 

If  A,  a,  b  are  given,  B  may  be  determined  from  the  relation 

(1)  AnB  =  bsmA- 

a 

If  log  sin  B  =  0,  the  triangle  is  a  right  triangle.     Why  ? 

If  log  sin  B  >  0,  the  triangle  is  impossible.     Why  ? 

If  log  sin  B  <  0,  there  are  two  possible  values,  Bu  B2  of  5, 
which  are  supplementary. 

Hence  there  may  be  two  solutions  of  the  triangle.  (See 
Example.) 

No  confusion  need  arise  from  the  various  possibilities  if  the 
corresponding  figure  is  constructed  and  kept  in  mind. 

It  is  desirable  to  go  through  the  computation  for  log  sin  B 

*  A  small  discrepancy  in  the  last  figure  need  not  cause  concern.    Why  ? 


66 


PLANE  TRIGONOMETRY 


[VI,  §  49 


before  making  out  the  rest  of  the  blank  form,  unless  the  data 
obviously  show  what  the  conditions  of  the  problem  actually 


Example  L     Given  :  A  =  46°  22'.2,  a  =  1.4063,  b  =  2.1048.    (Fig.  38.) 
To  find:  B,  C,  c. 

Formula :  sin  B  =  bsinA . 


Fig.  38 

Numbers  Logarithms 

6  =  2.1048  (->)  0.32321 

sin  A  =  sin  46°  22'  .2  (->)   (  +  )  9.85962-10 
bsinA  0.18283 

a  =1.4063  (->)   (-)  0.14808 

sin  B  (-<-)  0.03475 

Hence  the  triangle  is  impossible.     Why  ? 

Example  2.     Given :  a  =  73.221,  b  =  101.53,  A  =  40°  22'.3.    (Fig.  39.) 
To  find :  B,  C,  c. 

Formula:  sin£=&sin^. 


Numbers  Logarithms 

b  =  101.53  (->*)  2.00660 

sin  ^L=  sin  40°  22'. 3  (->)   (  +  )     9.81140  -  10 


6sin^i 

a  =  73.221 
sin  i? 


11.81800-10 
(->*)   (-)     1.86464 

9.95336  -  10 

The  triangle  is  therefore  possible  and 
has  two  solutions  (as  the  figure  shows) . 
We  then  proceed  with  the  solution  as 
follows  : 

We  find  one  value  2?i  of  B  from 
the  value  of  log  sin  B.  The  other 
value  B2  of  B  is  then  given  by  B2  = 
180°  -  Bx. 


VI,  §  49]  LOGARITHMIC  COMPUTATION  67 


Other  formulas : 

C=  180°  -(A  +  B). 

a  sin  C 
sin  A 

Check:   ^^ 

c  +  b 

_tanKC-B)< 
tan£(C  +  B) 

Numbers 

Logarithms 

sin  B 

9.95336  -  10 

i*i=    63°  55'. 2 

179°  60\0 

B2  =  116°    4' .8 

A  +  Bx  =  104°  17'.5 

179°  60'.0 

d=    75°42'.5 

a 

(->)            1.86464 

sin  .4 

(-►)   (-)  9.81140-10 

a/ahiA 

2.05324 

sin  d  =  sin  75°  42'. 5  (-►)   (  +  )  9.98634  -  10 
d  =  109.54  O-)  2.03958 

d-b=      8.01  (->)  0.90363 

ci  +  6  =  211.07  (->)  (-)  2.32443 

8.57920  -  10 
Cl-Bl=    11°47'.3 
Ci  +  Bi  =  139°  37 '.7 
tan  4(Ci—  JBi)=  tan  5°  53'. 6  (-►)     9.01377  -  10 

tan  K  Ci  +  -Bi)  =  tan  69°  48' .  8        (-*►)     0.43455 

8.57922  -  10 


\  Check. 


One  solution  of  the  triangle  gives,  therefore,  B  =  63°  55'. 2,  C  =  75°  42'. 5, 
c  =  109.54. 

To  obtain  the  second  solution,  we  begin  with  B2  =  116°  4'. 8.  We  find 
C2  from  C2  =  180°  -  (A  +  B2);  i.e.  C2  =  23°  32'. 9.  The  rest  of  the  com- 
putation is  similar  to  that  above  and  is  left  as  an  exercise. 


EXERCISES 

1.    Show  that,  given  J.,  a,  6,  if  A  is  obtuse,  or  if  J.  is  acute  and  a  >  6, 
there  cannot  be  more  than  one  solution. 

Solve  the  following  triangles  and  check  the  solutions  : 
J 2.   a  =  32.479,     6  =  40.176,     A  =  37°  25M. 


68 


PLANE  TRIGONOMETRY 


[VI,  §  49 


/: 


3.  6  =  4168.2, 

4.  a  =  2.4621, 

5.  a  =  421.6, 

6.  a  =  461.5, 


3179.8, 
4.1347, 
532.7, 


c  =  121.2, 


B  =  51°21'A. 
B  =  101°  37'.3. 
A  =  49°  21 '.8. 
C=22°31'.6. 


7.    Find  the  areas  of  the  triangles  in  Exs.  2-5. 

50.   Case  III.     Given  Two  Sides  and  the  Included  Angle. 

Example.     Given:  a=214.17,  6=356.21, 

B  C  =  62°  21 '.4.     (Fig.  40.) 


/  N. 

To  find:  A,  B,  c. 

V       ^v 

Formulas : 

V                                  \ 

tan| 

(B-A)=lL=Jtt^l(B  +  A); 

p>        \ 

B  +  A  =  180°  -  O  =  117°  38'.6 
a  sin  C 

C                             b  =  356.Sl                           J. 

Fig.  40 

sin  J. 

Numbers 

Logarithms 

6  -  a  =  142.04 

c-*o 

2.15241 

6  +  a  =  570.38 

(-» 

(-)  2.75616 

(6  -  a)/(b  +  a) 

9.39625  -  10 

tan |(1?  +  A)  =  tan 58°  49'.3 

(-» 

(  +  )  0.21817 

tan^(£-  A)=  tan  22°  22'. 2 

(«-) 

9.61442  -  10 

.-.  J.=        36°27'.l 

.Ans. 

2*=        81°  11'.5 

J.W8. 

a  =  214.17 

(— ►) 

2.33076 

sin^L  =  sin36°27'.l 

(-» 

(-)  9.77389-  10 

a/sin  .4 

2.55687 

sin  C  =  sin  62°  21'.  4 

(-» 

(  +  )  9.94736-10 

c  =  319.32  .4ns. 

(«-) 

2.50423 

Check  by  finding  log  (6/sin  B). 

I 

SXERCI 

SES                             f 

Solve  and  check  each  of  the  following  triangles  : 

1.  a  =  74.801,     6  =  37.502,     C  =  63°35'.5. 

^  2.  a  =  423.84,     6  =  350.11,     G  =  43°  14'.7. 

-s    3.  6  =  275,  c  =  315,  A  =  30°  30/. 

4.  a  =  150.17,     c  =  251.09,     B  =  40°40'.2; 

>  6.  a  =  0.25089,  6  =  0.30007,   C  =  42°  30'  20". 

6.  Find  the  areas  of  the  triangles  in  Exs.  1-5. 


VI,  §  51]         LOGARITHMIC  COMPUTATION 


69 


51.   Case 

IV.     Given  the 

Sides. 

Example. 

Given:  a  =  261.62, 

6  =  322.42, 

c  =  291.48. 

To  find:  A,  B,  C. 

Formulas  : 

s  =  K« 

+  b  +  c). 

r_J(« 

—  a)  0  —  6)  (8  —  c) 

r-yj 

8 

tan  i  A  =     r 

-,    tani£  =  -^-, 

s- 

a                      «  —  6 

Check  :  A  +  B  +  C  =  180°. 

Numbers 

a  =  261.62 

6  =  322.42 

c  =  291.48 

< 

28  =  875.52 

8  =  437.76 

8  — 

a  =  176.14 

8  - 

b  =  115.34 

8- 

-  c  =  146.28 

tan  \C- 


s  —  c 


Logarithms 


(-►) 


s  =  437.76  (Check). *(- 


2.24586 

2.06198 
)     (  +  )  2.16518 

6.47302 
)     (-)  2.64124 

3.83178 


r 

s1  —  a 
tan  |  A  -  tan  25°  4'. 1 

r 
s-b 
tan£  Bz=  tan35°32'.4 

r  = 
s  —  c  = 


(«-) 


«-) 


(«-) 


1.91589 
2.24586 
9.67003  -  10 

1.91589 
2.06198 
9.85391  -  10 

191589 
2.16518 
9.75071  -  10 


A=    50°    8'.2  Ans. 

B=    71°    4'.8  ^?is. 

C  =    58°  46'.9  ^Ins. 

179°  59'.9       (Check.) 


"7Vys> 


*By  adding  8—  a,  8  —  6,  s 

(A-*)* 


r^-t  (ft-  e}«  t^.    ^  fc  A 


(§37) 


70  PLANE  TRIGONOMETRY  [VI,  §  51 

EXERCISES 

Solve  and  check  each  of  the  following  triangles  : 
VI.    a  as  2.4169,    b  =  3.2417,    c  =  4.6293. 
*%!.<*=  21.637,    &  =  10.429,    c  =  14.221. 

5.  a  as  528.62,  .  6  =  499.82,    c  =  321.77. 
4.    a  =  2179.1,    6  =  3467.0,    c  =  5061.8. 

V«    a  =  0.1214,    &  =  0.0961,    c  =  0.1573. 

6.  Find  the  areas  of  the  triangles  in  Exs.  1-5. 

7.  Find  the  areas  of  the  inscribed  circles  of  the  triangles  in  Exs.  1-5. 

OTHER   LOGARITHMIC    COMPUTATIONS 
52.   Interest  and  Annuities. 

Simple  Interest. 

Let  the  principal  be  represented  by  P 

the  interest  on  $  1  for  one  year  by  r 

the  number  of  years  by  n 

the  amount  of  P  for  n  years  by  An 

Then  the  simple  interest  on  P  for  a  year  is  Pr 

the  amount  of  P  for  a  year  is  P  +  Pr  =P  (1-4-  r), 

the  simple  interest  on  P  for  n  years  is  Pnr 

the  amount  of  P  for  n  years  is  An  =P(1  +  nr). 

Example.     How  long  will  it  take  $210,   at  4%  simple  interest,  to 
amount  to  $  298.20  ? 

An  =  P(l  +  nr)  i.e.  n  =  An~  P. 

Pr 

Number  Logarithm 

An  -  P  =  88.20  ->-  1.9455 

Pr=    8.40  -^  0.9243 

n  =  10.5  -«—  1.0212     10  yr.  6  mo.  ^Ins. 

Compound  Interest. 
Let  the  original  principal  be  P 

and  the  rate  of  interest  r 

Then  the  amount  A]  at  the  end  of  the  first  year  is 


Ax  =  P-hPr=:P(l-\-r), 


VI,  §  52]  LOGARITHMIC  COMPUTATION  71/ 

the  amount  A2  at  the  end  of  the  second  year  is 

A2  =  A1(l  +  r)  =  P(l  +  ry, 
the  amount  at  the  end  of  n  years  is 

4,«J»(l+r)". 
If  the  interest  is  compounded  semiannually,  An-—  pf  1  +  M   , 

1+-)   ,  if  q  times  a  year^ln=P(  1  +  -  j   ■ 

Since  P  in  n  years  will  amount  to  AH,  it  is  evident  that  P  at 
the  present  time  may  be  considered  as  equivalent  in  value  to 
A  due  at  the  end  of  n  years.  Hence  P  is  called  the  present 
worth  of  a  given  future  sum  A.     Since 

An  =  P(l  +  r)%    P=  An (1  +  r)"\ 

Example.  In  how  many  years  will  one  dollar  double  itself  at  4  %  in- 
terest compounded  annually  ? 

An  =  P(\  +  r)-  or  log  ^  =  nlog(l  +  r). 

.    n  =  logA-log-P 
log  (1  +  r) 

Hence  n  =  log2  -  log  1  =  0,3010  =  17.7. 

log  (1.04)        0.0170 

17  yr.  9  mo.  Ans. 

Annuities.  An  annuity  is  a  fixed  sum  of  money  payable 
at  equal  intervals  of  time. 

To  find  the  present  worth  of  an  annuity  of  A  dollars  pay- 
able annually  for  n  years,  beginning  one  year  hence,  the  rate 
of  interest  being  r  and  the  number  of  years  n. 

Since  the  present  worth  of  the  first  payment  is  A  (1  +  r)_1, 
of  the  second  A(l  -f-  r)-2,  etc.,  the  present  worth  of  the  whole 
is 

P=^[(l  +  r)-i+(l-f  r)-*+  .-  +(l  +  r)-*]. 

The  quantity  in  the  brackets  is  a  G.  P.  whose  ratio  is  (1  +  r)~K 
Summing,  we  have 

l-(l  +  r)-i  r\_        {1  +  ryj 


72  PLANE  TRIGONOMETRY  [VI,  §  52 

If  the  annuity  is  perpetual,  i.e.  n  is  infinite,  the  formula  for 

A 

present  worth  becomes  P  -—  —  • 

Example.     What  should  be  paid  for  an  annuity  of  $  100  payable  an- 
nually for  20  years,  money  being  worth  4  %  per  annum  ? 


p=Mh LLl. 

0.04  L         (1.04) 20 J 

020  =  2.188. 

Therefore  P=  —  fl L-1  =2500  f  U^§1  =$1358,  approximately. 

0.04  L       2.188  J  L2.188J  '    FF  J 


(1.04) 
By  logarithms  (1 .04)  20  -  2. 188. 


53.  Projectiles.  Logarithms  are  used  extensively  in  ballis- 
tic computations.  [Ballistics  is  the  science  of  the  motion  of 
a  projectile.]  The  following  is  a  very  simple  example  of  the 
type  of  problem  considered. 

The  time  of  flight  of  a  projectile  (in  vacuum)  is  given  by 

the  formula  T=\- *  where  X  is  the  horizontal  range 

*        9 
in  feet,  <f>  is  the  angle  of  departure,  and  g  is  the  acceleration 
due  to  gravity  in  feet  per  second  per  second  \_g  —  32.2].     If  it 
is  known  that  the  range  is  3000  yd.  and  that  the  angle  of  de- 
parture is  30°  20',  find  the  time  of  flight. 


T        /2Xtan<£ 
"X         9 

Numbers 

Logarithms 

21=  18000 

~* 

4.2553 

tan  30°  20' 

-* 

9.7673  -  10 
4.0226 

32.2 

"* 

1.5079. 

2)2.5147 

18.09 

<— 

1.2574     T  =  18.09  seconds. 

Ans. 

EXERCISES 

1.  Find  the  amount  of  $  500  in  10  years  at  4  per  cent  compound  inter- 
est, compounded  semiannually. 

2.  In  how  many  years  will  a  sum  of  money  double  itself  at  5  per  cent 
interest  compounded  annually  ?  semiannually  ? 


VI,  §  54]  LOGARITHMIC  COMPUTATION  73 

3.  A  thermometer  bulb  at  a  temperature  of  20°  C.  is  exposed  to  the  air 
for  15  seconds,  in  which  time  the  temperature  drops  4  degrees.  If  the 
law  of  cooling  is  given  by  the  formula  0  =  doe-61,  where  6  is  the  final  tem- 
perature, #o  the  initial  temperature,  e  the  natural  base  of  logarithms,  and 
t  the  time  in  seconds,  find  the  value  of  b. 

4.  The  stretch  s  of  a  brass  wire  when  a  weight  m  is  hung  at  its  free 

end  is  given  by  the  formula  j 

8  =  — — , 

where  m  is  the  weight  applied  in  grams,  g  =  980,  I  is  the  length  of  the 
wire  in  centimeters,  r  is  the  radius  of  the  wire  in  centimeters,  and  fc  is  a 
constant.  If  m  =  844.9  grams,  I  =  200.9  centimeters,  r  =  0.30  centi- 
meter when  s  =  0.056,  find  k. 

5.  The  crushing  weight  P  in  pounds  of  a  wrought-iron  column  is  given 

by  the  formula  ,73.55 

P=  299,600^—, 
p 

where  d  is  the  diameter  in  inches  and  I  is  the  length  in  feet.  What  weight 
will  crush  a  wrought-iron  column  10  feet  long  and  2.7  inches  in  diameter? 

6.  The  number  n  of  vibrations  per  second  made  by  a  stretched  string 
is  given  by  the  relation  2     rz-r- 

n  =  2TV^r' 

where  I  is  the  length  of  the  string  in  centimeters,  M  is  the  weight  in 
grams  that  stretches  the  string,  m  the  weight  in  grams  of  one  centimeter 
of  the  string,  and  g  =  980.  Find  n  when  M  =  5467.9  grams,  I  =  78.5 
centimeters,  m  =  0.0065  gram. 

7.  The  time  t  of  oscillation  of  a  pendulum  of  length  I  centimeters  is 
given  by  the  formula  ,— — 

>(980 
Find  the  time  of  oscillation  of  a  pendulum  73.27  centimeters  in  length. 

8.  The  weight  w  in  grams  of  a  cubic  meter  of  aqueous  vapor  saturated 
at  17°  C.  is  given  by  the  formula 

=     1293  x  12.7  x  5 

(1  +  ^X760x8)* 
Compute  w. 

54.  The  Logarithmic  Scale.  An  arithmetic  scale  in  which  the 
segments  from  the  origin  are  proportional  to  the  logarithms  of  1,  2,  3,  etc., 
is  called  a  logarithmic  scale.     Such  a  scale  is  given  in  Fig.  42. 


i      I     JIIIJ1 

Fig.  42 


74 


PLANE  TRIGONOMETRY 


[VI,  §  55 


55.    The  Slide  Rule.      The  slide  rule  consists  of  a  rule  along  the 
center  of  which  a  slip  of  the  same  material  slides  in  a  groove.     Along  the 


Fig.  43 

upper  edge  of  the  groove  are  engraved  two  logarithmic  scales,  A  and  B, 
that  are  identical.  Along  the  lower  edge  are  also  two  identical  logarithmic 
scales,  0  and  D,  in  which  the  unit  is  twice  that  in  scales  A  and  B.  Since 
the  segments  represent  the  logarithms  of  the  numbers  found  in  the  scale, 
the  operation  of   adding  the  segments  is  equivalent  to  multiplying  the 


f 

1                            2                £ 

* 
1 

4        I 

)      6 

7    i 

5    9    1 

2 

A      1    !    1    1   Ml  ii     i  I  I  I 

! 

Ml  1 1  mil  ilmlilililililililili 

iiilii 

1,1,1  1 

J  EH 

III 

IIJJ  11,1,1 ,1, 

jTI'II 

WMV-, 

\ 

r          oL 

1 

II 

IK 

II 

II  III 

But 

1 

JIIIIJII 

\ 

B  r 

\ 

!'     !        i 

\ 

Dl 

2 

3^         4 

5       6      7    8    9 

) 

rl                ■     ■ 

2 

3 

/ 

I             C- 

1  1  ll  Hill 

llllll    II 

M'lll 

Ii  IjlJ  ll 

tiin 

TtT&U 

n,  INI  INI  IIIIIHIIHI  HI 

nil 

I  If 

MM  1 

J.-l 

1.1  . 

1Jftf4 

.njijl 

1 

2 

3 

4 

Fig.  44 

corresponding  numbers.  Thus  in  Fig.  44  the  point  marked  1  on  scale  B 
is  set  opposite  the  point  marked  2.5  on  scale  A.  The  point  marked  4  on 
scale  B  will  be  opposite  the  point  marked  10  on  scale  A,  i.e.  2.5  x  4  =  10. 
Similarly  we  read  2.5  x  3.2  =  8,  2.5  x  2.5  =  6.25.  Other  multiplications 
can  be  performed  in  an  analogous  manner. 

Division'can  be  performed  by  reversing  the  operation.  Thus  in  Fig.  44 
every  number  of  scale  B  is  the  result  of  dividing  the  number  above  it  by 
2.5.    Thus  we  read  7.2  -~  2.5  =  2.9  approximately. 

Since  scales  G  and  D  are  twice  as  large  as  scales  A  and  B,  it  follows 
that  the  numbers  in  these  scales  are  the  square  roots  of  the  numbers 
opposite  to  them  in  scales  A  and  B.  Conversely  the  numbers  on  scales 
A  and  B  are  the  squares  of  the  numbers  opposite  them  on  scales  C  and 
D.  Moreover  the  scales  C  and  D  can  be  used  for  multiplying  and  divid- 
ing, but  the  range  of  numbers  is  not  so  large. 

For  a  more  complete  discussion  of  the  use  of  a  slide  rule  consult  the 
book  of  instructions  published  by  any  of  the  manufacturers  of  slide  rules, 
where  also  exercises  will  be  found  for  practice. 


CHAPTEK   VII 
TRIGONOMETRIC  RELATIONS 

56.  Radian  Measure.  In  certain  kinds  of  work  it  is  more 
convenient  in  measuring  angles  to  use,  instead  of  the  degree, 
a  unit  called  the  radian.  A  radian  is  defined  as  the  angle  at 
the  center  of  a  circle  whose  subtended  arc  is  equal  in  length 
to  the  radius  of  the  circle  (Fig.  45).  Therefore,  if  an  angle  $ 
at  the  center  of  a  circle  of  radius  r  units  subtends  an  arc  of 
s  units,  the  measure  of  6  in  radians  is 

r 

Since  the  length  of  the  whole  circle  is  2  -n-r,  it  follows  that 

—  =  2tt  radians  =  360°, 
r 

or 

(2)  it  radians  =  180°. 

Therefore, 

180° 

TT 


1  radian  = =  57°  17'  45"  (approximately).  FlG  45 


It  is  important  to  note  that  the  radian  *  as  defined  is  a  con- 
stant angle,  i.e.  it  is  the  same  for  all  circles,  and  can  therefore 
be  used  as  a  unit  of  measure. 

From  relation  (2)  it  follows  that  to  convert  radians  into 
degrees  it  is  only  necessary  to  multiply  the  number  of  radians 
by  180/7T,  wliile  to  convert  degrees  into  radians  we  multiply 
the  number  of  degrees  by  tt/180.  Thus  45°  is  tt/4  radians  ; 
7r/2  radians  is  90°. 

*  The  symbol  r  is  often  used  to  denote  radians.  Thus  2r  stands  for  2 
radians,  irr  for  tt  radians,  etc.  When  the  angle  is  expressed  in  terms  of  it  (the 
radian  being  the  unit),  it  is  customary  to  omit  r.  Thus,  when  we  refer  to  an 
angle  it,  we  mean  an  angle  of  it  radians.  When  the  word  radian  is  omitted, 
it  should  be  mentally  supplied  in  order  to  avoid  the  error  of  supposing  ir 
means  180.     Here,  as  in  geometry,  t  =  3.14159.  .  .  . 

75 


76  PLANE  TRIGONOMETRY  [VII,  §  57 

57.   The  Length  of  Arc  of  a  Circle.     From  relation  (1), 
§  56,  it  follows  that 

s  =  r8. 

That  is  (Fig.  46),  if  a  central  angle  is  measured 
in  radians,  and  if  its  intercepted  arc  and  the 
radius  of   the  circle  are  measured  in  terms  of 
the  same  unit,  then 
length  of  arc  =  radius  x  central  angle  in  radians. 

r~  EXERCISES 

1.   Express  the  following  angles  in  radians  : 

25°,  145°,  225°,  300°,  270°,  450°,  1150°. 

-*   2.   Express  in  degrees  the  following  angles  : 

■K               7  IT         blT         0             5-TT 
—  ,      — ,       ,      u  7T, . 

4'  6         6  '4 

*  3.  A  circle  has  a  radius  of  20  inches.  How  many  radians  are  there  in 
an  angle  at  the  center  subtended  by  an  arc  of  25  inches  ?  How  many 
degrees  are  there  in  this  same  angle  ?  Ans.  |r ;  71°  37'  approx. 

— i  4.  Find  the  radius  of  a  circle  in  which  an  arc  12  inches  long  subtends 
an  angle  of  35°. 

"""  5.  The  minute  hand  of  a  clock  is  4  feet  long.  How  far  does  its  ex- 
tremity move  in  22  minutes  ? 

6.  In  how  many  hours  is  a  point  on  the  equator  carried  by  the  rotation 
of  the  earth  on  its  axis  through  a  distance  equal  to  the  diameter  of  the  earth? 

7.  A  train  is  traveling  at  the  rate  of  10  miles  per  hour  on  a  curve  of 
half  a  mile  radius.     Through  what  angle  has  it  turned  in  one  minute  ? 

8.  A  wheel  10  inches  in  diameter  is  belted  to  a  wheel  3  inches  in 
diameter.     If  the  first  wheel  rotates  at  the  rate  of  5  revolutions  per  \\g 
minute,  at  what  rate  is  the  second  rotating?     How  fast  must  the  former 
rotate  in  order  to  produce  6000  revolutions  per  minute  in  the  latter  ? 

58.   Angular    Measurement    in    Artillery    Service.      The 

divided  circles  by  means  of  which  the  guns  of  the  United  States  Field 
Artillery  are  aimed  are  graduated  neither  in  degrees  nor  in  radians,  but 
in  units  called  mils.     The  mil  is  defined  as  an  angle  subtended  by  an  arc 
of  ^^-q  of  the  circumference,  and  is  therefore  equal  to 
2tt       3.1416 


6400   3200 


0.00098175  =(0.001  -  0.00001825)  radian. 


VII,  §  58]        TRIGONOMETRIC  RELATIONS 


77 


The  mil  is  therefore  approximately  one  thousandth  of  a  radian. 
(Hence  its  name.)* 

Since  (§57) 
length  of  arc  =  radius  x  central  angle  in  radians, 
it  follows  that  we  have  approximately 

length  of  arc  = x  central  angle  in  mils  ; 

1000 

i.e.  length  of  arc  in  yards  a  (radius  in  thousands  of  yards)  •  (angle 
in  mils).     The  error  here  is  about  2  % . 

Example  1.  A  battery  occupies  a  front  of  60  yd.  If  it  is 
at  5500  yd.  range,  what  angle  does  it  subtend  (Fig.  47)?  We 
have,  evidently, 

angle  =  —=  11  mils. 
5.5 


Example  2.  Indirect  Fire,  t  A  battery 
posted  with  its  right  gun  at  G  is  to  open  fire  on 
a  battery  at  a  point  T,  distant  2000  yd.  and  in- 
visible from  G  (Fig.  48).  The  officer  directing 
tfie  fire  takes  post  at  a  point  B  from  which  both 
the  target  T  and  a  church  spire  P,  distant 
3000  yd.  from  <?,  are  visible.  B  is  100  yd.  at 
the  right  of  the  line  6?  T  and  120  yd.  at  the 
right  of  the  line  GP  and  the  officer  finds  by 
measurement  that  the  angle  PBT  contains 
3145  mils.  In  order  to  train  the  gun  on  the 
P  target  the  gunner  must  set  off  the  angle  PG  T 
on  the  sight  of  the  piece  and  then  move  the  gun 


Fig.  48 


*  To  give  an  idea  of  the  value  in  mils  of  certain  angles  the  following  has 
been  taken  from  the  Drill  Regulations  for  Field  Artillery  (1911),  p.  164: 

"  Hold  the  hand  vertically,  palm  outward,  arm  fully  extended  to  the  front. 
Then  the  angle  subtended  by  the 

width  of  thumb  is 40  mils 

width  of  first  finger  at  second  joint  is     .        ;        .        .        .40  mils 
width  of  second  finger  at  second  joint  is         ....      40  mils 

width  of  third  finger  at  second  joint  is 35  mils 

width  of  little  finger  at  second  joint  is 30  mils 

width  of  first,  second,  and  third  fingers  at  second  joint  is  .  115  mils 
These  are  average  values." 
,  t  The  limits  of  the  text  preclude  giving  more  than  a  single  illustration  of 
the  problems  arising  in  artillery  practice.  For  other  problems  the  student  is 
referred  to  the  Drill  Regulations  for  Field  Artillery  (1911) ,  pp.  57, 61, 150-164 ; 
and  to  Andrews,  Fundamentals  of  Military  Service,  pp.  153-159,  from  which 
latter  text  the  above  example  is  taken. 


78  PLANE  TRIGONOMETRY  [VII,  §  58 

until  the  spire  P  is  visible  through  the  sight.     When  this  is  effected,  the 
gun  is  aimed  at  T. 

Let  F  and  E  be  the  feet  of  the  perpendiculars  from  B  to  GT  and  GP 
respectively,  and  let  B  T'  and  BP'  be  the  parallels  to  G  T  and  GP  that 
pass  through  B.  Then,  evidently,  if  the  officer  at  B  measures  the  angle 
PBT,  which  would  be  used  instead  of  angle  PG  T  were  the  gun  at  B  in- 
stead of  at  G,  and  determines  the  angles  TBT'  =  FTB  and  PBP1  =  EPB, 
he  can  find  the  angle  PG  T  from  the  relation 

PGT  =  PBT  =  PBT-  TBV-PBP*. 

Now  tan  FTB  =  — ,  tan  EPB  =  — . 

TF  PE 


small  compared  with  G  T  and  GP  respectively,  the  radian  measure  of  the 
angle  is  approximately  equal  to  the  tangent  of  the  angle.  Why  ?  Hence 
we  have 

FB) 


FTB  =  tan  FTB 

GT 

EPB  =  tan  EPB  =  — 
GP 


approximately. 


Therefore       TBT'  =  FTB  =  —  radians  =  50  mils, 

2000 

PBP1  =  EPB  =  i^-  radians  =  40  mils. 
3000 

Hence  PGT  =  PBT  -  TBT'  -  PBP1 

=  3145  -  50  -  40 

=  3055  mils, 

which  is  the  angle  to  be  set  off  on  the  sight  of  the  gun. 

Hence  from  the  situation  indicated  in  Fig.  48  we  have  the  following 

rule  : 

(1)  Measure  in  mils  the  angle  PBT  from  the  aiming  point  P  to  the 
target  T  as  seen  at  B. 

(2)  Measure  or  estimate  the  offsets  FB  and  EB  in  yards,  the  range 
G  T  and  the  distance  GP  of  the  aiming  point  P  in  thousands  of  yards. 

(3)  Compute  in  mils  the  offset  angles  by  means  of  the  relations 

TBT'  =  FTB, 
PBP'  =  EPB, 

TBT'  =  ^B~- 
GT 

PBP'  =  —  • 
GP 

(4)  Then  the   angle   of   deflection  PGT  is  equal  to  the  angle  PBT 
diminished  by  the  sum  of  the  offset  angles. 


VII,  §  59]         TRIGONOMETRIC  RELATIONS  79 

EXERCISES 

1.  A  battery  occupies  a  front  of  80  yd.  It  is  at  5000  yd.  range. 
What  angle  does  it  subtend  ? 

2.  In  Fig.  48  suppose  PBT  =  3000  mils,  FB  =  200  yd.,  G  T  =  3000  yd., 
EB  =  150  yd.,  GP  =  4000  yd.     Find  the  number  of  mils  in  PG  T. 

3.  A  battery  at  a  point  G  is  ordered  to  take  a  masked  position  and  be 
ready  to  fire  on  an  indicated  hostile  battery  at  a  point  T  whose  range  is 
known  to  be  2100  yd.  The  battery  commander  finds  an  observing  station 
B,  200  yd.  at  the  right  and  on  the  prolongation  of  the  battery  front,  and 
175  yd.  at  the  right  of  PG.  An  aiming  point  P,  5900  yd.  in  the  rear,  is 
found,  and  PBT  is  found  to  be  2600  mils.     Find  PG  T. 

4.  A  battery  at  a  point  G  is  to  fire  on  an  invisible  object  at  a  point  T 
whose  range  is  known  to  be  2000  yd.  A  battery  commander  finds  an 
observing  station  B,  100  yd.  at  the  right  of  G  T  and  150  yd.  at  the  right 
of  GP.  The  aiming  point  P  is  1500  yd.  in  front  and  to  the  left  of  G  T. 
The  angle  TBP  contains  1200  mils.     Find  PG  T. 

59.   The  Sine  Function.     Let  us  trace  in  a  general  way  the 
variation  of  the  function  sin  6  as  6  increases  from  0°  to  360°. 
For  this  purpose  it  will  be  convenient  to  think  of  the  distance 
r  as  constant,  from  which  it   follows   that 
the  locus  of  P  is  a  circle.     When  6  =  0°,  the 
point   P  lies  on  the  #-axis  and  hence  the 
ordinate   is   0,  i.e.  sin  0°  =  0/r  =  0.      As  6 
increases    to    90°,    the    ordinate    increases 
until  90°  is  reached,  when  it  becomes  equal 
to  r.      Therefore,  sin  90°  =  r/r  =  1.      As  0  FlG  49 

increases  from  90°  to  180°,  the  ordinate  de- 
creases until  180°  is  reached,  when  it  becomes  0.  Therefore 
sin  180°  =  0/r  =  As  $  increases  from  180°  to  270°,  the  ordi- 
nate of  P  continually  decreases  algebraically  and  reaches  its 
smallest  algebraic  value  when  0  =  270°.  In  this  position  the 
ordinate  is  —  r  and  sin  270°  =  —  r/r  =  —  1.  When  0  enters 
the  fourth  quadrant,  the  ordinate  of  P  increases  (algebraically) 
until  the  angle  reaches  360°,  when  the  ordinate  becomes  0. 


80 


PLANE  TRIGONOMETRY 


[VII,  §  59 


Hence,  sin  360°  =  0.     It  then  appears  that : 

as  6  increases  from  0°  to  90°,  sin  0  increases  from  0  to  1 ; 

as  0  increases  from  90°  to  180°,  sin  6  decreases  from  1  to  0 ; 

as  0  increases  from  180°  to  270°,  sin  0  decreases  from  0  to  —  1 ; 

as  6  increases  from  270°  to  360°,  sin  6  increases  from  —  1  to  0. 
It  is  evident  that  the  function  sin  6  repeats  its  values  in  the 
same  order  no  matter  how  many  times  the  point  P  moves 
around  the  circle.  We  express  this  fact  by  saying  that  the 
function  sin  6  is  periodic  and  has  a  period  of  360°.  In  symbols 
this  is  expressed  by  the  equation 

sin  [8  +  n  •  360°]  =  sin  9, 

where  «  is  any  positive  or  negative  integer. 

The  variation  of  the  function  sin  6  is  well  shown  by  its 
graph.  To  construct  this  graph  proceed  as  follows :  Take  a 
system  of  rectangular  axes  and  construct  a  circle  of  unit  radius 


Fig.  50 


with  its  center  on  the  #-axis  (Fig.  50).  Let  angle  XM4P  =  0. 
Then  the  values  of  sin  6  for  certain  values  of  6  are  shown  in 
the  unit  circle  as  the  ordinates  of  the  end  of  the  radius  drawn 
at  an  angle  6. 


e 

0 

30° 

45° 

60° 

90° 

sin  0 

0 

MiPt 

MtPt 

MZPZ 

M4P4 

... 

Now  let  the  number  of  degrees  in  0  be  represented  by  dis- 
tances measured  along  OX.  At  a  distance  that  represents  30° 
erect  a  perpendicular  equal  in  length  to  sin  30° ;  at  a  distance 


VII,  §  60]        TRIGONOMETRIC  RELATIONS 


81 


that  represents  60°  erect  one  equal  in  length  to  sin  60°,  etc. 
Through  the  points  0,  Pl9  P2,  —  draw  a  smooth  curve ;  this 
curve  is  the  graph  of  the  function  sin  0. 

If  from  any  point  P  on  this  graph  a  perpendicular  PQ  is 
drawn  to  the  ic-axis,  then  QP  represents  the  sine  of  the  angle 
represented  by  the  segment  OQ. 

Since  the  function  is  periodic,  the  complete  graph  extends 
indefinitely  in  both  directions  from  the  origin  (Fig.  51). 


1&*X 


ilar  to  those 


60.   The  Cosine  Function.     By  arguments  s 
used  in  the  case  of  the  sine  function  we  may  show  that : 
as  8  increases  from  0°  to  90°,  the  cos  6  decreases  from  1  to  0 ; 
as  0  increases  from  90°  to  180°,  the  cos  0  decreases  from  0  to  —  1 ; 
as  6  increases  from  180°  to  270°,  the  cos  0  increases  from  —  1  to  0 ; 
as  6  increases  from  270°  to  360°,  the  cos  0  increases  from  0  to  1. 

The  graph  of  the  function  is  readily  constructed  by  a  method 


Fig.  52 


similar  to  that  used  in  the  case  of  the  sine  function.     This  is 
illustrated  in  Fig.  52. 

The  complete  graph  of  the  cosine  function,  like  that  of  the 
sine  function,  will  extend  indefinitely  from  the  origin  in  both 


82 


PLANE  TRIGONOMETRY 


[VII,  §  60 


directions  (Fig.  53).     Moreover  cos  6,  like  sin  6,  is  periodic  and 
has  a  period  of  360°,  i.e. 

COS  [6  4-  71  •  360°]  as  cos  6, 
where  n  is  any  positive  or  negative  integer. 

Y 


61.  The  Tangent  Function.  In  order  to  trace  the  varia- 
tion of  the  tangent  function,  consider  a  circle  of  unit  radius 
with^its  center  at  the  origin  of  a  system  of  rectangular  axes 
(Fig.  54).  Then  construct  the  tangent  to 
this  circle  at  the  point  M(l,  0)  and  let  P 
denote  any  point  on  this  tangent  line.  If 
angle  MOP  =  0,  we  have  tan  6  =  MP/OM 
ae  MP/1  =  MP,  i.e.  the  line  MP  represents 
tan0. 

Now  when  $  =  0°,  MP  is  0,  i.e.  tan  0°  is  0. 
As  the  angle  6  increases,  tan  6  increases.  As 
0  approaches  90°  as  a  limit,  MP  becomes 
infinite,  i.e.  tan  6  becomes  larger  than  any  number  whatever. 

At  90°  the  tangent  is  undefined.     It  is  sometimes  convenient 
to  express  this  fact  by  writing 

tan  90°  =oo. 

However  we  must  remember  that  this  is  not  a  definition  for 
tan  90°,  for  oo  is  not  a  number.  This  is  merely  a  short  way  of 
saying  that  as  0  approaches  90°  tan  0  becomes  infinite  and 
that  at  90°  tan  0  is  undefined. 

Thus  far  we  have  assumed  0  to  be  an  acute  angle  approach- 
ing 90°  as  a  limit.  Now  let  us  start  with  0  as  an  obtuse  angle 


Fig.  54 


VII,  §  61]         TRIGONOMETRIC  RELATIONS 


83 


and  let  it  decrease  towards  90°  as  a  limit.  In  Fig.  55  the  line 
MP'  (which  is  here  negative  in  direction)  represents  tan  0. 
Arguing  precisely  as  we  did  before,  it  is 
seen  that  as  the  angle  0  approaches  90° 
as  a  limit,  tan  6  again  increases  in  magni- 
tude beyond  all  bounds,  i.e.  becomes  infi- 
nite, remaining,  however,  always  negative. 
We  then  have  the  following  results. 

(1)  When  0  is  acute  and  increases  to- 
wards 90°  as  a  limit,  tan  0  always  remains 
positive  but  becomes  infinite.     At  90°  tan  0  is  undefined. 

(2)  When  0  is  obtuse  and  decreases  towards  90°  as  a  limit, 
tan  6  always  remains  negative  but  becomes  infinite.  At  90° 
tan  6  is  undefined. 

It  is  left  as  an  exercise  to  finish  tracing  the  variation  of  the 
tangent  function  as  6  varies  from  90°  to  360°.  Note  that 
tan  270°,  like  tan  90°,  is  undefined.  In  fact  tan  n  •  90°  is  unde- 
fined, if  n  is  any  odd  integer. 


Fig. 


Fig.  56 


To  construct  the  graph  of  the  function  tan  6  we  proceed 
along  lines  similar  to  those  used  in  constructing  the  graph  of 
sin  6  and  cos  0.  The  following  table  together  with  Fig.  56 
illustrates  the  method. 


84 


PLANE  TRIGONOMETRY 


[VII,  §  61 


e 

0° 

30° 

45° 

60° 

90° 

120° 

135° 

150° 

180° 

210° 

tan  0 

0 

MPX 

MP2 

MPZ 

undefined 

MPA 

MPb 

MP6 

ilfP7=0 

MPX 

It  is  important  to  notice  that  tan  0,  like  sin  6  and  cos  0,  is 
periodic,  but  its  period  is  180°.     That  is 

tan(e  +  n-180o)=tan6, 

where  w  is  any  positive  or  negative  integer. 


X 


EXERCISES 

1.  What  is  meant  by  the  period  of  a  trigonometric  function  ? 

2.  What  is  the  period  of  sin  0  ?  cos  0  ?  tan  0  ? 

3.  Is  sin  0  defined  for  all  angles  ?  cos  0  ? 

4.  Explain  why  tan  0  is  undefined  for  certain  angles.     Name  four 
angles  for  which  it  is  undefined.     Are  there  any  others  ? 

5.  Is  sin  (0  +  360°)  =  sin  0  ? 

6.  Is  sin  (0  +  180°)  =  sin  0  ? 

7.  Is  tan  ( 0  +  180°)  =  tan  0  ? 

8.  Is  tan  (0  +  360°)  =  tan'0  ? 

Draw  the  graphs  of  the  following  functions  and  explain  how  from  the 
graph  you  can  tell  the  period  of  the  function  : 

9.  sin0.  11.   tan0.  13.    sec0. 
10.    cos0.                                  12.    csc0.  14.'  ctn0. 

Verify  the  following  statements  : 

15.  sin90°  +  sin270°  =  0.  18.  cos  180°  +  sin  180°  =-  1. 

16.  cos  90°  +  sin0°  =  0.  19.  tan  360°  +  cos  360°  =  1. 
tan  1 80°  +  cos  1 80°  =  -  1 .  20.  cos 90° + tan  180°-Isin270^ = 1.^ 

21.    Draw  the  graphs  of  the  functions  sin  0,  cos  0,  tan  0,  making  use  of 
a  table  of  natural  functions.     See  p.  112. 
\2fc)  Draw  the  curves  y  =  2 sin 0  ;  y  =  2  cos 0  ;  y  =  2  tan  6. 

23.  Draw  the  curve  y  =  sin  0  +  cos  0. 

24.  From  the  graphs  determine  values  of  0  for  which  sin  0  =  \  ;  sin  0 
=  1  ;  tan  0  =  1;  cos  0  =  \  ;  cos  0  =  1. 


VII,  §  63]         TRIGONOMETRIC  RELATIONS 


85 


62.  The  Trigonometric  Functions  of  —  9.  Draw  the  angles 
6  and  —  0,  where  OP  is  the  terminal  line  of  0  and  OP  is  the 
terminal  line  of  —  6.    Figure  57  shows  an  angle  6  in  each  of 

r 


Fig  57 


the  four  quadrants.     We  shall  choose  OP  =  OP  and  («,  y)  as 

the  coordinates  of  P  and  (x',  y')  as  the  coordinates  of  P'.     In 

all  four  figures 

t!  =»  x,  y'  =  -  y,  r'  =  r. 
Hence 

sin(-0)  =  ^  =  :^  =  -sin0, 
r  r 


cos  ( —  6)  m  —  ==  -  =  cos  6, 
r'      r 


_?/ 


y — 


tan  (  -  0)  =  2-  =  —a  =  -  tan  (9. 


Also, 


esc  ( —  6)  =  —  esc  6  ;  sec  ( —  0)  =  sec  0  ;  ctn  (  —  6)  =  —  ctn  0. 
The  above  results  can  be  stated  as  follows :  The  functions  of 
—  6  equal  numerically  the  like  named  functions  of  6.     The 
algebraic  sign,  however,  will  be  opposite  except  for  the  cosine 
and  secant. 

Example,    sin- 10°  = -sin  10°,  cos- 10°  =  cos  10°,  tan-10°=  -tan  10°. 

63.   The  Trigonometric  Functions  of  180°  +  6.     Similarly, 
the  following  relations  hold  : 

sin  (180°  +  0)  =  —  sin  0,  esc  (180°  +  6)  =  -  esc  0, 

cos  (180°  +  6)  =  -  cos  0,  sec  (180°  +  6)  =  -  sec  6, 

tan  (180°  +  6)  =  tan  0,  ctn  (180°  +  6)  =  ctn  0. 

The  proof  is  left  as  an  exercise. 


86/  PLANE  TRIGONOMETRY  [VII,  §  64 

64.  Summary.  An  inspection  of  the  results  of  §§  27-28, 
62-63  shows : 

1.  Each  f miction  of  —  0  or  180°  ±  0  is  equal  in  absolute  value 
(but  not  always  in  sign)  to  the  same  function  of  0. 

2.  Each  function  of  90°  —  0  is  equal  in  magnitude  and  in  sign 
to  the  corresponding  co-function  of  6. 

These  principles  enable  us  to  find  the  value  of  any  function 
of  any  angle  in  terms  of  a  function  of  a  positive  acute  angle 
(not  greater  than  45°  if  desired)  as  the  following  examples 
show. 

Example  1.     Reduce  cos  200°  to  a  function  of  an  angle  less  than  45°. 

Since  200°  is  in  the  third  quadrant,  cos  200°  is  negative.  Hence 
cos  200°  =  -  cos  20°.     Why  ? 

Example  2.     Reduce  tan  260°  to  a  function  of  an  angle  less  than  45°. 

Since  260°  is  in  the  third  quadrant,  tan  260°  is  positive.  Hence 
tan  260°  =  tan  80°  =  ctn  10°  (§  27). 

Example  3.  Reduce  sin  (—  210°)  to  a  function  of  a  positive  angle 
less  than  45°. 

From  §  62  we  know  sin  —  210°  =  —  sin  210°. 

Considering  the  positive  angle  210°,  we  have 

sin  -  210°  =  -  sin  210°  =  -  [  -  sin  30°]  =  sin  30°. 

EXERCISES 

Reduce  to  a  function  of  an  angle  not  greater  than  45°  : 

1.  sin  163°.  5.    esc  901°. 

2.  cos  (-110°).  *"+  i.    ctn  (-1215°).    +  | 

Ans.    -sin 20°.  7>    tan 840°. 

->   3.   sec  (-265°).  8.    sin  510°. 

4.   tan  428°.  tX—  tv. 

Eind  without  the  use  of  tables  the  values  of  the  following  functions  : 
— >9.    cos  570°.  11.    tan  390°.  13.    cos  150°. 

10.'   sin  330°.  ^*  12.    sin  420°.  14.    tan  300°. 

Reduce  the  following  to  functions  of  positive  acute  angles  : 
^15.   sin  250°.  T*  18.    sec  (-245°). 

Ans.    —  sin  70°  or  —  cos  20°.  19.    Csc(—  321°). 

16.  cos  158°.  20.   sin  269°. 

17.  tan  (-389°). 


VII,  §  64]         TRIGONOMETRIC  RELATIONS  87 

Prove  the  following  relations  from  a  figure  : 


(a)     sin  (90°  +  0)  =  cos  0. 

(O 

sin  (180°  +  0)  =  — sini 

cos  (90°  +  0)  =  —  sin  0. 

cos  (180°+  0)  =  -cos 

tan  (90°  +  0)  =  —  ctn  0. 

tan  (180°  +  0)=  tan0. 

esc  (90°  +  0)=sec0. 

csc(18O°  +  0)  =  —  csci 

sec  (90°  +  0)  =  -  esc  0. 

sec  (180°  +  0)  =  -seci 

ctn  (90°  +  0)  =  -  tan  0. 

ctn  (180°  +  0)=ctn0. 

(b)  sin (180°-  6)=sm0. 

(<*) 

sin  (270°  -0)  =-cos 

cos  (180°  —  6)  =  —  cos  6. 

cos  (270° —0)  =  - sin  i 

tan  (180°  -  0)  =  -tan0. 

tan  (270°  -  0)  =  ctn  0. 

esc  (180°  —  d)  =  esc  6. 

esc  (270°  —  0)  =  —  sec 

sec  (180°  —  d)  =  —  sec0. 

sec  (270°  -  0)  =  -  esc 

ctn  (180°  -0)  =  -  ctn  6. 

ctn  (270°  -  0)  =  tan  0. 

(e)   sin  (270°  +  0)  =  -  cos  0. 
cos  (270°  +  0)  =  sin  0. 
tan  (270°  +  0)  =  -  ctn  0. 
esc  (270°  +  0)  =  -  sec  0. 
sec  (270°  +  0)  =  esc  0. 
ctn  (270°  +  0)  =  -  tan  0. 


tJ4t^r 


Hn  ik 
a  o 

B 


e  ta        csa 


— 


fcuk 


MlljJIlMllH 


CHAPTER    VIII 


TRIGONOMETRIC   RELATIONS  (Continued) 

^5.  Trigonometric  Equations.  An  identity,  as  we  have 
seen  (§  26),  is  an  equality  between  two  expressions  which  is 
satisfied  for  all  values  of  the  variables  for  which  both  expres- 
sions are  defined.  If  the  equality  is  not  satisfied  for  all 
values  of  the  variables  for  which  each  side  is  defined,  it  is 
called  a  conditional  equality,  or  simply  an  equation.  Thus 
1  —  cos  0  =  0  is  true  only  if  0  =  n  •  360°,  where  n  is  an  integer. 
To  solve  a  trigonometric  equation,  i.e.  to  find  the  values  of  0 
for  which  the  equality  is  true,  we  usually  proceed  as  follows. 

1.  Express  all  the  trigonometric  functions  involved  in  terms 
of  one  trigonometric  function  of  the  same  angle. 

2.  Find  the  value  (or  values)  of  this  function  by  ordinary 
algebraic  methods. 

3.  Eind  the  angles  between  0°  and  360°  which  correspond  to 
the  values  found.     These  angles  are  called  particular  solutions. 

4.  Give  the  general  solution  by  adding  n  •  360°,  where  n  is 
any  integer,  to  the  particular  solutions. 


Example  1.     Find  6  when  sin  6  =  $. 
The  particular  solutions  are  30°  and  150°. 
30°  +  n  ■  360°,  150°  +  n  •  360°. 


The  general  solutions  are 


Example  2.     Solve  the  equation  tan  6  sin  d  —  sin  0  =  0. 

Factoring  the  expression,  we  have  sin  0  (tan  6  —  Y)=  0.      Hence  we 
have  sin  0  =  0,  or  tan  6  —  1  =  0.     Why  ? 

The  particular  solutions  are  therefore  0°,  180°,  45°,  225°.     The  genera! 
solutions  are  n  .  360°,  180°  +  n  .  360°,  45°  +  n  •  360°,  225°  +  n  •  360°. 

88 


2. 

sin  0  = — —  • 
2  , 

3. 

2 

4. 

2 

5. 

tan0  =  —  1. 

JL 

ctn  0=1. 

16. 

2  sin  0  =  tan  0. 

VIII,  §  66]       TRIGONOMETRIC  RELATIONS  89 

EXERCISES 

Give  the  particular  and  the  general  solutions  of  the  following 
equations : 

tJq  7.    sec  0  —  2. 

2  8.  tan  0  =  0. 

Vi  9.  sec2  0  =  2. 

10.  sin20  =  |. 

11.  cos0=  —  £. 

12.  csc20  =  f 
/l3.  4  sin  0  —  3  esc  0  =  0. 
1 14.  2  sin  0  cos2  0  =  sin  0. 

)l5.    cos  0  -f  sec  0  =  f . 
^Irw.    Particular  solutions  :  0°,  180°,  60°,  300°. 
17.   3  sin  0  +  2  cos  0  =  2.  /l8.    2  cos2  0—1  =  1  —  sin2  0. 

Inverse  Trigonometric  Functions.     The  equation 

x  —  sin  y  (1) 

may  be  read : 

y  is  an  angle  whose  sine  is  equal  to  x, 

a  statement  which  is  usually  written  in  the  contracted  form 

y  =  arc  sin  x.*  (2) 

For  example,  x  =  sin  30°  means  that  x  =  \,  while  y  =  arc  sin  i 

means  that  y  =  30°,  150°,  or  in  general  (n  being  an  integer), 

30°  +  n  •  360°  ;  150°  +  n  •  360°. 

Since  the  sine  is  never  greater  than  1  and  never  less  than 

—  1,  it  follows  that  —l_\x—\l.     It  is  evident  that  there  is 

an  unlimited  number  of  values  ofy  =  arc  sin  x  for  a  given  value 

of  x  in  this  interval. 

We  shall  now  define  the  principal  value  Arc  sin  x  f  of  arc  sin  x, 

distinguished  from  arc  sin  x  by  the  use  of  the  capital  A,  to  be 

*  Sometimes  written  y  =  sin-1  x.  Here  —  1  is  not  an  algebraic  exponent, 
but  merely  a  part  of  a  functional  symbol.  When  we  wish  to  raise  sin  x  to 
the  power  —  1,  we  write  (sin  x)-}. 

t  Sometimes  written  Sin-i  x,  distinguished  from  sin-1  x  by  the  use  of  the 
capital  S. 


90 


PLANE  TRIGONOMETRY 


[VIII,  §  66 


the  numerically  smallest  angle  whose  sine  is  equal  to  x.  This  func- 
tion like  arc  sin  x  is  denned  only  for  those  values  of  x  for 
which 

The  difference  between  arc  sin  x  and  Arc  sin  x  is  well  illus- 
trated by  means  of  their  graph.  It  is 
evident  that  the  graph  ofy  =  arc  sin  x, 
i.e.  x  =  sin  y  is  simply  the  sine  curve 
with  the  role  of  the  x  and  y  axes  inter- 
changed. (See  Fig.  58.)  Then  for  every 
admissible  value  of  x,  there  is  an  un- 
limited number  of  values  of  y ;  namely, 
the  ordinates  of  all  the  points  P1}  P2,  •-,  in 
which  a  line  at  a  distance  x  and  parallel 
to  the  2/-axis  intersects  the  curve.  The 
single-valued  function  Arc  sin  x  is  repre- 
sented by  the  part  of  the  graph  between 
M  and  N, 
Similarly  arc  cos  x,  defined  as  "  an  angle  whose  cosine  is  x," 

has  an   unlimited'  number   of  values  for 

every  admissible  value  of  x(—  1  f^  x  <  1) 

We  shall  define  the  principal  value  Arc 

cos  x  as  the  smallest  positive  angle  whose 

cosine  is  x.     That  is. 


Fig.  58 


0  ^  Arc  cos  x  <^  7r. 

Figure  59  represents  the  graph  of  y  =  arc 
cos  x,  and  the  portion  of  this  graph  between 
M  and  N  represents  Arc  cos  x. 

Similarly  we  write  x  =  tan  y  as  y  =  arc 
tan  x,  and  in  the  same  way  we  define  the 
symbols  arc  ctn  x ;   arc   sec  x ;    arc  esc  x. 
The  principal  values  of  all  the  inverse  trigonometric  functions 
are  given  in  the  following  table. 


Y 

2tt 

37T 

P» 

N\ 

IT 

7T 

2 

ft 

M 

-1 

0 

Pi 

1  X 

V=  arc  cos  x 
y=Arc  cos  x 
Fig.  59 


VIII,  §  66]       TRIGONOMETRIC  RELATIONS 


91 


y- 

Arc  sin  x 

Arc  cos  x 

Arc  tan  x 

Range  of  x 

-lgx^ 1 

-l^a^l 

all  real  values 

Range  of  y 

7T     .         7T 

to  — 

2        2 

0  tO    7T 

to  — 

2        2 

x  positive 

1st  Quad. 

1st  Quad. 

1st  Quad. 

x  negative 

4th  Quad. 

2d  Quad. 

4th  Quad. 

Arc  ctn  x 

Arc  sec  x 

Arc  cscx 

Range  of  x 

all  values 

x^l  orx^-  1 

a;^lorx2-l 

Range  of  y 

OtOT 

0  tO  7T 

to  — 

2        2 

x  positive 

1st  Quad. 

1st  Quad. 

1st  Quad. 

x  negative 

2d  Quad. 

2d  Quad. 

4th  Quad. 

In  so  far  as  is  possible  we  select  the  principal  value  of  each 
inverse  function,  and  its  range,  so  that  the  function  is  single- 
valued,  continuous,  and  takes  on  all  possible  values.  This  ob- 
viously cannot  be  done  for  the  Arc  sec  x  and  for  Arc  esc  y. 

EXERCISES 

1.   Explain  the  difference  between  arc  sin  x  and  Arc  sin  x. 
>^.   Find  the  values  of  the  following  expressions  : 
\(a)  Arc  sin  \.  (d)  Arc  tan  —  1. 


(e)  arc  cos 


V3 


(/)  Arc  cos  22. 


>>^&)  arc  sin  \. 

(c)  arc  tan  1.  2 

S^What  is  meant  by  the  angle  it  ?    tt/4  ? 

4.   Through  how  many  radians  does  the  minute  hand  of  a  watch  turn 
in  30, minutes  ?  in  one  hour  ?  in  one  and  one  half  hours  ? 


6.   For  what  values  of  x  are  the  following  functions  defined  : 
y\d)  arc  sin  x  ?  ^/($)  arc  tan  x  ?  _--^e)  arc  sec  x  ? 

(6)  arc  cos  x  ?  (d)  arc  ctn  x  ?  (/)  arc  esc  x  ? 

6.    What  is  the  range  of  values  of  the  functions  : 
(or)  Arc.  sin  x  ?  (c)  Arc  tan  x  ?  (e)  Arc  sec  x. 

(6)  Arc  cos  x  ?  (d)  Arc  ctn  x  ?  (/)  Arc  esc  x  ? 


fa  I 


<0 


TW 


92 


PLANE  TRIGONOMETRY 


[VIII,  §  66 


7.  Draw  the  graph  of  the  functions  : 

(a)  arc  sin  x.  (c)  arc  tan  x.  (e)  arc  sec  x. 

(&-)  arc  cos x.  (d)  arc  etna;.  (/)  arc  esc x. 

8.  Find  the  value  of  cos  (Arc  tan  f). 

Hint.     Let  Arc  tan  f  =  6.    Then  tan  d  =  £  and  we  wish  to  find   the 
value  of  cos  6. 

:9.    Find  the  values  of  cos  (arc  tan  f ) .     !^V>    • 
TIC    Find  the  value  of  the  following  expressions  : 
(a)  sin  (arc  cos  |).     *"*  -(c)  cos  (Arc  cos  T5^).        (e)  sin  (Arc  sin  \). 
(6)  sin  (arc  sec  3).     {$)  sec  (Arc  esc  2).  (/)  tan  (Arc  tan  5) . 

11.  Prove  that  Arc  sin  (2/5)=  Arc  tan  (2/V21) 

12.  Find  x  when  Arc  cos  (2  x2  -  2  x)  =  2  tP/3.  \  ^-  0 
Find  the  values  of  the  following  expressions  : 


13.    cos  [90°— Arc  tan f]. 


j£f 1^1 


f 


14.  sec  [90°  — Arc  sec  2].- 

15.  tan  [90°  -  Arc  sin  T\]. 

67.  Projection.  Consider  two  directed  lines  p  and  q  in  a 
plane,  i.e.  two  lines  on  each  of  which,  one  of  the  directions 
has  been  specified  as  positive  (Fig.  60).  Let  A  and  B  be 
any  two  points  on  p  and  let  A',  B'  be  the  points  in  which  per- 


Fig.  00 


pendiculars  to  q  through  A  and  B,  respectively,  meet  q.  The 
directed  segment  A'B'  is  called  the  projection  of  the  directed  seg- 
ment AB  on  q  and  is  denoted  by 

A'B'  =  projff  AB. 
In  both  figures  AB  is  positive.     In  the  first  figure  A'B'  is  posi 
tive,  while  in  the  second  figure  it  is  negative. 

As  special  cases  of  this  definition  we  note  the  following : 


VIII,  §  67]       TRIGONOMETRIC  RELATIONS  93 

1.  If  p  and  q  are  parallel  and  are  directed  in  the  same  way, 

we  have 

proj,  AB  =  AB. 

2.  If  p  and  q  are  parallel  and  are  directed  oppositely,  we 

have 

projff  AB  =  —  AB. 

3.  If  p  is  perpendicular  to  q,  we  have 

proj,  AB  =  0. 
It  should  be  noted  carefully  that  these  propositions  arc  true 

no  matter  how  A  and  B  are  situated  on  p. 

We  may  now  prove  the   following  important  proposition  : 
If  A  and  B  are  any  two  points  on  a  directed  line  p,  and  q  is 

any  directed  line  in  the  same  plane  with  p,  then  we  have  both 

in  magnitude  and  sign 

(1)  projg  AB  =  AB  ■  cos  (pq)*  =  AB  .  cos  (qp). 

We  note  first  from  §  8  that  (pq)+  (qp)  =  0  +  n-  360°,  where 
n  is  any  integer.  Hence  from  §  64,  cos  (jxj)  =  cos  (qp).  Two 
cases  arise. 


Jt 

T22 


Fig.  61 

Case  1.  Suppose  AB  is  positive,  i.e.  it  has  the  same  direc- 
tion as  p. 

Through  A  draw  a  line  q^  parallel  to  q  and  with  the  same 
direction.  [It  is  evident  that  we  may  assume  without  loss  of 
generality  that  q  is  horizontal  and  is  directed  to  the  right.] 
Let  A'B'  be  the  projection  of  AB  on  q  and  let  BB'  meet  qx 
in  Bx.     Then  by  the  definition  of  the  cosine  we  have 

AB 

——±  =  cos  (qip)  =  cos  (pqi)  =  cos  (qp)  =  cos  ( pq) 
AB 

*  (pq)  represents  an  angle  through  which  p  may  be  rotated  in  order  to 
make  its  direction  coincide  with  the  direction  of  q  ;  similarly  for  (qp). 


94 


PLANE  TRIGONOMETRY 


[VIII,  §  67- 


in  magnitude  and  sign.     Hence 

AB±  =  AB  ■  cos  (pq)  =  AB  •  cos  (qp). 
But  ABX  =  A'B'  =  proj3  AB. 

Therefore      projtf  AB  =  AB  •  cos  (pp)  =  AB  •  cos  (qp). 

Case  2.     Suppose  AB  is  negative. 

If  AB  is  negative,  BA  is  positive  and  we  have  from  Case  1, 

B'A!  =  BA  •  cos  (pq)  =  BA  •  cos  (qp). 
Changing  the  signs  of  both  members  of  this  equation,  we  have 

A'B'  =  AB  •  cos  (})q)=  AB  •  cos  (qp). 

The  special  cases  1,  2,  3,  are  obtained  from  formula  (1) 
by  placing  (qp)  or  (pq)  equal  to  0°,  180°,  90°  respectively. 

Theorem.  If  A,  B,  C  are  any  three  points  in  a  plane,  and  I 
is  any  directed  line  in  the  plane,  the  algebraic  sum  of  the  projec- 
tions of  the  segments  AB  and-  BC  on  I  is  equal  to  the  projection 
of  the  segment  AC  on  I. 

As  a  point  traces  out  the  path  from  A  to  B,  and  then  from 
B  to  C  (Fig.  62),  the  projection  of  the  point  traces  out  the 
segments  from  A'  to  B'  and  then  from  B' 
to  C.  The  tjjjft  result  of  this  motion  is  a 
motion  from  A'  to  O  which  represents 
the  projection  of  AC,  i.e. 

A'B'  +  B'C  =  A' C. 


I 

/ 

— ' — ■ 

z^ 

B 

ti 

C 

>^ 

A' 

( 

i' 

h 

' 

EXERCISES 

1.  What  is  the  projection  of  a  line  segment  upon  a  line  I,  if  the  line 
segment  is  perpendicular  to  the  line  I  ? 

2.  Find  projx^4JB  and  proj^l?*  in  each  of  the  following  cases,  if  a 
denotes  the  angle  from  the  x-axis  to  AB. 

(a)  AB  =  5,  a  =  60°.  (c)  AB  =  6,  a  =  90°. 

(6)^45  =  10,  a  =  300°.  (d)  AB  =  20,  a  =  210°. 

*  Projx  AB  and  proj,,  AB  mean  the  projections  of  AB  on  the  x-axis  and 
the  y-axis,  respectively. 


<^  w- 


<-^ 


VIII,  §  68]       TRIGONOMETRIC  RELATIONS  95 

3.  Prove  by  means  of  projection  that  in  a  triangle  ABC 

a—b  cos  C  -f  c  cos  B. 

4.  If  projj.  AB  =  3  and  proj„  AB  =  —4,  find  the  length  of  AB. 

5.  A  steamer  is  going  northeast  20  miles  per  hour.  Hots  fast  is  it 
going  north  ?  going  east  ? 

6.  A  20  lb.  block  is  sliding  down  a  15°  incline.  Find  what  force 
acting  directly  up  the  plane  will  just  hold  the  block,  allowing  ope  half  a 
pound  for  friction. 

7.  Prove  that  if  the  sides  of  a  polygon  are  projected  in  order  upon  any 
given  line,  the  sum  of  these  projections  is  zero. 


Fig.  63 


The  Addition  Formulas.  We  may  now  derive  formulas 
for  sin  (a  -f-  /3),  cos  (a  -f-  ft),  and  tan  (a  +  ft)  in  terms  of  func- 
tions of  a  and  ft.  To  this  end 
let  P(x,  y)  be  any  point  on  the 
terminal  side  of  the  angle  a  (the 
initial  side  being  along  the  posi- 
tive end  of  the  a>axis  and  the 
vertex  being  at  the  origin).  The 
angle  a  +  ft  is  then  obtained  by 
rotating  OP  through  an  an^le 
ft.  If  P'  (x',  y')  is  the  new  Sta- 
tion P  after   this  rotation  and 

OP  =  OP'  =  r,  we  have  sin  (a  -f-  ft)  =  £ ,  cos  (a  +  ft)  =  - ,  by 

v  r 

definition.     Our  first  problem  is,  therefore,  to  find  x'  and  y'  in 

terms  of  x,  y,  and  ft. 

In  the  figure  OMP  is  the  new  position  of  the  triangle  OMP 
after  rotating  it  about  0  through  the  angle  ft.     Now, 

x'  =  projx  OP'  ss  projx  OM'  +  projx  M'F 

=  xcosft  +  ycos(ft  +  ^\ 

=  x  cos  ft  —  y  sin  ft. 


96  PLANE  TRIGONOMETRY  [VIII,  §  68 

Similarly, 

y>  =  proj,  OP'  =  proj,  OM'  +  projtf  M'F 

=  x  cos(?--  ft\+  y  cos  ft 

=  x  sin  ft  +  y  cos  /?. 

Hence,  ,     ,    ~x      y'      x ,-+  n  .  V         n 

'      sm  («  +  j3)=V-=-  sr$/3+^  cos  £ 


r      r 

=  sin  a  cojr  [ 


or  (1)  m       sin  (a  +  P)  =  sin  a  co#  p  -f  cos  a  sin  p. 


Also 


cos 


s(«  +  £)  =  ^-  =  -^osft-^  sin/?. 


or  (2)  cos  (a  +  p)  =  cos  a  cos  p  —  sin  a  sin  p. 

Further  we  have 

tan  (a  4-  B)  =  S*n  (a  ~*~  ®  =  sni  g  cos  ft  +  cos  <*  sin  ft 
cos  (a  4-  ft)      cos  a  cos  ft  —  sin  a  sin  /J 

Dividing  numerator  and  denominator  by  cos  a  cos  ft,  we  have 

(3)  •      tan(a+B)=  tan*  +  tanP. 
w  v        K;      1  -  tan  a  tan  p 

Furthermore,  by  replacing  ft  by  —  ft  in  (1),  (2),  and  (3),  and 
recalling  that 

sin  (—  ft)  =  —  sin  ft,  cos  (—  ft)  =  cos  ft,  tan  (—  ft)  =  —  tan  ft, 
we  obtain  -^^fc_ 

(4)  sin  (a  —  P)  =  sin  a  cmf$  —  cos  a  sin  p, 

(5)  cos  (a  —  P)  =  cos  a  cm$  ■+-  sin  a  sin  p, 


(6)  tan  (a.  -  tt  =   tan  o^  tan  p 


tan  (a  -  p)  =   —  y  —  r 
v        r/      1  +  t*n  a  tan  p 


EXERCISES 

Expand  the  iollowing  : 
-*-±r  sin  (45°  +  «)  =        3.    cos  (60°  +  a)  =  5.   sin  (30°  -  45°)  = 

—=«.   tan  (30°  -  0)  =        4.    tan  (45°  +  60°)  =       .  6.    cos  (180°  -  45°)  = 

7.    What  do  the  following  formulas  become  if  «  =  /3  ? 
sin  («  +  (S)=  sin  a  cos  /3  +  cos  a  sin  p.  t      (a  A-  8s)—  tan  a  +  tan  P  . 

sin  (a  —  /3)  =  sin  a  cos  /3  —  cos  a  sin  0.  *  1  —  tan  a  tan  /3 

cos  (a  +  /3)  =  cos  a  cos  0  —  sin  a  sin  /S.  .       ,     _  q\  _   tan  a  —  tan  g  ( 

cos  (a  —  /3)  =  cos  a  cos  p  +  sin  a  sin  j8.  1  +  tan  a  tan  /3 


VIII,  §  68]       TRIGONOMETRIC  RELATIONS  97 

8.    Complete  the  following  formulas  : 

sin  2  a  cos  a  +  cos  2  a  sin  a  —  tan  2  a  +  tan  a  _ 

sin  3  a  cos  a  —  cos  3  a  sin  a  =  1  —  tan  2  a  tan  a 

-^*   Prove  sin  75°  =  V^  +  1,  cos  75°  =  V^  ~  1 ,  tan75°  =  Vg  +  1- 
2V2  2V2  V3-1 

10.   Given  tan  a  =  f ,  sin  ft  =  T5^,  and  a  and  ft  both  positive  acute  angles, 
find  the  value  of  tan  (a  +  ft);  shr(a~—  ft);  cos  (a  +  ft);  tan  (a  —  ft). 

,-.  »r1 1.   Prove  that 

(a)  cos  (60°  +  a)  +  sin  (30°  +  a)  =  cos  a. 
*     ( 6)  sin  (60°  +  0)  -  sin  (60°  -  6)  -  sin  0. 

(c)  cos  (30°  +  0)-  cos  (30°  -  0)=  -  sin  6. 

(d)  cos  (45°  +  6)  +  cos  (45°  -  0)  =  V2  •  cos  0. 

>      (e)  sin  1  a  +  -  )  +  sin  (  a  —  —  j  =  sin  a. 
(/)  cos  ( a  +  - )  +  cos  (a  —  -)  =  V3  •  cos  a. 

~* — 12.   By  using  the  functions  of  60°  and  30°  find  the  value  of  sin  90°  ; 
cos  90°. 

13.  Find   in   radical  form  the  value   of  sin  15°  ;    cos  15°  ;    tan  15°  ; 
sin  105°  ;  cos  105°  ;  tan  105°. 

14.  If  tan  a  =  |,  sin  ft  =  T5T,  and  a  is  in  the  third  quadrant  while  ft  is 
in  the  second,  find  sin  (a  ±  ft) ;  cos  (a  ±  ft) ;  tan  (a  ±  ft). 

Prove  the  following  identities  :  <^"~" 

15     sin  (a  +  ft)  _  tan  a  +  tan  ft  _         16     sin  2  a  ,  cos  2  a  _  sm  3  a 

sin  (a  — ft)     tana  — tan  ft  sec  a        esc  a 

17     tana -tan  (a -ft)  =  tan  ^        19.  (a)  sin  (180o  _  9)  -  sin  $m 

1  +  tan  a  tan  (a— ft)  (6)  cos  (180°  -  6)  =  -  cos  0. 

x18.  tan(0±45°)  +  ctn(0T45°)=O.  (c)  tan  (180°  -  6)  =  -  tan  0. 

20.  cos  (a  -f  ft)  cos  (a  —  ft)  =  cos2  a  —  sin2  ft. 

21.  sin  (a  +  ft)  sin  (a  —  ft)  =  sin2  a  -  sin2  ft. 

22.  ctn(«  +  /9)  =  ctnttctng-1.        23.  ctn  (a  -  ft)  =  Ctn "  ctn**  +  *  . 

ctna  +  ctnft  ctnft  — etna 

24.  Prove  Arc  tan  £  +  Arc  tan  |  =  ?r/4. 

[Hint  :  Let  Arc  tan  \  =  x  and  Arc  tan  \  =  y.     Then  we  wish  to  prove 
x  +  y  =  ir/4,  which  is  true  since  tan  (x  +  y)=  1.] 

25.  Prove  Arc  sin  a  +  Arc  cos  a  =  -  if  0  <  a  <  1. 

p 

26.  Prove  Arc  sin  T*7  -f  Arc  sin  |  =  Arc  sin  ||. 

i  H 


A 


98 x  PLANE  TRIGONOMETRY  [VIII,  §  68 

27.  Prove  Arc  tan  2  +  Arc  tan  £  as  ir/2. 

28.  Prove  Arc  cos  §  +  Arc  cos  (—  T5y)  =  Arc  cos  (—  f|). 

29.  Prove  Arc  tan  T85  +  Arc  tan  f  =  Arc  tan  f  £. 
-30.  Find  the  value  of  sin  [Arc  sin  |  +  Arc  ctnf  ]. 
-  31.  Find  the  value  of  sin  [Arc  sin  a  +  Arc  sin  6]  if  0  <  a  <  1,  0  <  b  <  1. 

32.  Expand  sin  (x  +  y  +  z)  ;  cos(x  +  y  +  z). 
[Hint  :  x  +  y  +  z  =(x  +  y)+  z.] 

33.  The  area  i  of  a  triangle  was  computed  from  the  formula 
A  =  I  ab  sin  0.  If  an  error  c  was  made  in  measuring  the  angle  0,  show  that 
the  corrected  area  A'  is  given  by  the  relation.^.'  =  A(cos  e  +  sin  e  ctn  6). 

69.  Functions  of  Double  Angles.  In  this  and  the  follow- 
ing articles  (§§  69-71)  we  shall  derive  from  the  addition 
formulas  a  variety  of  other  relations  which  are  serviceable  in 
transforming  trigonometric  expressions.  Since  the  formulas 
for  sin  (a  +  fi)  and  cos  (a  +  /?)  are  true  for  all  angles  a  and  (3, 
they  will  be  true  when  /?  =  a.     Putting  /3  =  a,  we  obtain 

(1)  sin  2  a  =  2  sin  a  cos  a, 

(2)  cos  2  a  =  cos2  a  —  sin2  a. 
Since  sin2  a  +  cos2  a  —  1,  we  have  also 

(3)  cos  2  a  =  1  -  2  sin2  a 

(4)  =2cos2a-l. 

Similarly  the  formula  for  tan  (a  +  ft)  (which  is  true  for  all 
angles  a,  ft,  and  a+ft  which  have  tangents)  becomes,  when  ft=a, 

(5)  tan2q=     2tana    , 
v  ;  l-tan2a 

which  holds  for  every  angle  for  which  both  members  are  denned. 
The  above  formulas  should  be  learned  in  words.  For  ex- 
ample, formula  (1)  states  that  the  sine  of  any  angle  equals 
twice  the  sine  of  half  the  angle  times  the  cosine  of  half  the 
angle.     Thus  sin6^  =  2  sin3«  cos3^, 

2  tan  2  x 


tan  4  x  — 


l-tan22x' 


cos  x  =  cos2  -  —  sin2  -> 


VIII,  §  70]       TRIGONOMETRIC  RELATIONS  99 

70.   Functions  of  Half  Angles.     From  (3),  §  69,  we  have 


Therefore 

2sin2£=l  —  cos  a. 

(6) 

«in«_  |      /I -cos  a 

°m2_±\        2 

From  (4),  §  69,  we  have 


2  cos2  -  =  1  -+-  cos  a. 


Therefore 


(7)  cos«  =  ±/±f5^. 

Formulas  (6)  and  (7). are  at  once  seen  to  ! 
«.     Now,  if  we  divide  formula  (6)  by  formula  (7),  we  obtain 

/QX  *      a  /l  —  cos  a 

(8)  tan  -  =  ±  \/- , 

v  }  2       '  V'l  +  cosa' 

which  is  true  for  all  angles  a  except  n  •  180°,  where  n  is  any- 
odd  integer. 

Example.     Given  sin^.  =—  3/5,  cos 4  negative  ;  find  sin  (A/2). 

Since  the  angle  A  is  in  the  third  quadrant,  A/2  is  in  the  second  or 
fourth  quadrant,  and  hence  sin  (A/2)  may  be  either  positive  or  negative. 
Therefore,  since  cos  A  =  —  4/5,  we  have 


2  \     2  «/Tn  10 


VTo       io 


EXERCISES 

Complete  the  following  formulas  and  state  whether  they  are  true  for 
all  angles  : 

1.  sin  2  a  =  3-   tan  2  a  —  5.   cos "  = 

A 

2.  cos2a=     (three  forms).  4    sin-=  6.   tan  -  = 

2  2 

7.  In  what  quadrant  is  0/2  if  6  is  positive,  less  than  360°,  and  in  the 
second  quadrant  ?  third  quadrant  ?  fourth  quadrant  ? 

8.  Express  cos  2  a  in  terms  of  cos  4  a. 

9.  Express  sin  6  x  in  terms  of  functions  of  3  x. 


100  PLANE  TRIGONOMETRY  [VIII,  §  70 

10.  Express  tan  4  a  in  terms  of  tan  2  a. 

11.  Express  tan  4  a  in  terms  of  cos  8  a. 

12.  Express  sin  x  in  terms  of  functions  of  x/2. 

13.  Explain  why  the  formulas  for  sin  x  and  cos  x  in  terms  of  functions 
of  2  x  have  a  double  sign. 

14.  From  the  functions  of  30°  find  those  of  60°. 

15.  From  the  functions  of  60°  find  those  of  30°. 

16.  From  the  functions  of  30°  find  those  of  15°. 

17.  From  the  functions  of  15°  find  those  of  7°.  5. 

18.  Find  the  functions  of  2  a  if  sin  a  =  $  and  a  is  in  the  second 
quadrant. 

19.  Find  the  functions  of  a/2  if  cos  a  =—  0.6  and  a  is  in  the  third 
quadrant,  positive,  and  less  than  360°. 

20.  Express  sin  3  a  in  terms  of  sin  a.     [Hint  :  3a  =  2a  +  a.] 

21.  From  the  value  of  cos  45°  find  the  functions  of  22°.  5. 

22.  Given  sin  a  =  —  and  a  in  the  second  quadrant.    Find  the  values  of 
(a)  sin  2  a.  (c)   cos  2  a.  (e)   tan  2  a. 

(6)  sin".  (d)  cos?.  (/)  tan|. 

23.  If  tan  2  a  =  |  find  sin  a,  cos  a,  tan  a  if  a  is  an  angle  in  the  third 
quadrant. 

Prove  the  following  identities  : 

24.  1  +  C0*«=cto&  27.    l-cos2fl  +  sin2fl=tan0 
. sin  a                2  .      .  1  +  cos  2  0  +  sin  2  0 


25. 
26. 


Tsin  —  cos-]   =1  —  sin0.  28.    sin-  +  cos  —  =  ±  Vl  +  sina. 

L.2  2J  22 

cos2  0  +  cos0  4-l„ctn,,j  29    Be0a  +  tan«=ten^  +  ^V 

sin20  +  sin0  \4      2/ 

30.    2  Arc  cos  a;  =  Arc  cos  (2  x2— 1). 


31.   2  Arc  cosx  =  Arc  sin  (2  xVl  —  x2). 


32.   tan  [2  Arc  tanx]  =  ^^-.        34.    tan  [2  Arc  sec x]  =  ±  2 ' 


1  -  x2  J  2  -  x2 

33.    cos  [2  Arc  tan  x]  =     —  x  •       /35^  *os  (2  Arc  sin  a)  =  1  —  2  a2. 


1  +x2 
Solve  the  following  equations 

36.  cos  2  x  +  5  sin  x  =  3.  40.  sin2  2  x  —  sin2  x  ss  f . 

37.  cos2x  —  sinx  =  \.  41.  sin2x  =  2cosx. 

38.  sin  2  x  cos  x  =  sin  x.  42.  2  sin2  2  x  =  1  —  cos2x. 

39.  2sin2x  +  sin22x  =  2.  43.  ctnx  —  csc2x  —  1. 


fa 


VIII,  §  71]       TRIGONOMETRIC  RELATIONS  101 

44.  A  flagpole  50  ft.  high  stands  on  a  tower  49  ft.  high.  At  what  dis- 
tance from  the  foot  of  the  tower  will  the  flagpole  and  the  tower  subtend 
equal  angles  ? 

45.  The  dial  of  a  town  clock  h^s  a  diameter  (j|f  JO  ft.  and  its  center  is 
100  ft.  above  the  ground.     At1  what' distance  from  the  foot  of  the  tower 
will  the  dial  be 
must  be  as  large 


most  plainly  viqbie  f]  £r  he  ajn^'fubWr-ded  by  the  dial 
as  possible.]*  °  '  ••••  ' 


71.   Product  Formulas.     From  §  68  we  have 

sin  (a-\-  (3)  =  sin  a  cos  /?  -f  cos  a  sin  /3, 
sin  (a  —  /?)  =  sin  a  cos  /?  —  cos  a  sin  /?. 

Adding,  we  get 

(1)  sin  (a  +  p)  +  sin  (a  —  /?)  =  2  sin  a  cos  /3. 

Subtracting,  we  have 

(2)  sin  (a  +  ft)  —  sin  («  —  p)  =  2  cos  a  sin  0. 

Now,  if  we  let  a  -f-  /?  =  P  and  a  —  ft  =  Q, 
thell  «  =  ^,   0  =  Z^$. 

Therefore  formulas  (1)  and  (2)  become 

P  -+-  O         P 

sin  P  +  sin  Q  =  2  sin      ^  v  cos  — 


2  2 

Pi     Q  p 

sin  P  —  sin  Q  =  2  cos  — — *  sin  — 


2 

Similarly,  starting  with  cos  («  +  /?)  and  cos  (a  —  /?)  and  per- 
forming the  same  operations,  the  following  formulas  result : 

P  4-  O        P  —  O 
cos  P  +  cos  Q  =  2  cos  — —-*-  cos  — —i-, 

A  A 

cos  P  —  cos  Q  =  —  2  sin       J~  v  sin  — —^. 

2  2 


y^.  In  words : 
the  sum  of  two  sines  = 

twice  sin  (half  sum)  times  cos  (half  difference), 
the  difference  of  two  sines  = 

twice  cos  (half  sum)  times  sin  (half  difference),* 
*  The  difference  is  taken,  first  angle  minus  the  second. 


102  PLANE  TRIGONOMETRY  [VIII,  §  71 

the  sum  of  two  cosines  = 

twice  cos  (half  sum)  times  cos  (half  difference), 
the  difference  of  two  cosines,  = 

minus  twice  sin  (half  sum)  times  sin  (half  difference). * 
Example  1.  -Prove  that  '  a        f 

coB8a;  +  co8:a?=ctnja, 
sin  3  x  +  sin  x 
for  all  angles  for  which  both  members  are  defined. 

cos  3  x  +  cos  x_  2  cos  ^(3  x  4-  x)  cos  |(3  x  —  x)  _  cos  2  x  _    .    9 
sin  3  x  +  sin  x      2  sin  £(3  x  +  x)  cos  \  (3  x  —  x)  ~"  sin  2  x  ~~ 

Example  2.     Reduce  sin  4  x  4-  cos  2  x  to  the  form  of  a  product. 
We  may  write  this  as  sin  4  x  4-  sin  (90°  —  2x),  which  is  equal  to 
2  sin  Ix  +  W-Z*  cos  tx-W  +  az     2  sin  (45„  +  x)  cos  (3  x  _  45„}_ 

EXERCISES 

Reduce  to  a  product : 

1.  sin  4  0  —  sin  2  0.         4.    cos  2  0  +  sin  2  0.  7.    cos  3  x  +  sin  5  x. 

2.  cos  0  +  cos  3  0.  5.    cos  3  0  —  cos  6  0.  8.    sin  20°  —  sin  60°. 

3.  cos  6  0  +  cos  2  0.  6.    sin  (x  -f  Ax)  —  sin  x. 
Show  that 

9.    sin  20°  +  sin  40°  =  cos  10°.  12     sin  15°  4-  sin  75°  _  _        6QO 

10.  cos  50°  4- cos  70°  =  cos  10°.  '    sin  15°  -  sin  75°  ~ 

11.  sin75°-sinl5°  =  tan 30°.  13.     sin3  0-sin5  0  =  _  ^ 4 , 
cos  75°  4-  cos  15°  cos  3  0  —  cos  5  0 

Prove  the  following  identities  :  ,        ^ " 


"^ sm~$TT4r'Slfi~3  ft  _  g^fl"  15     sin  a  +  sin  ft  _  tan  \  (a  +  ft) 

cos  3  a  —  cos  4  a  2  '    sin  a  —  sin  ft     tan  £  (a  —  ft) 

. ..       cos  a  4-  2  cos  3  a  4-  cos  5  a       cos  3  a 

id. = • 

cos  3  a.  +  2  cos  5  a  4-  cos  7  a      cos  5  a 

-_     cos  a—  cos  ft  _  _  tan  ^(#4-  ft)         lg     sin  (n  —  2)  0  4-  sin  nd  _         . 
cos  «4- cos  ft         ctn£(a  —  ft)  '    cos  (n  —  2)  0  —  cos  nd 

Solve  the  following  equations  : 
■    19.    cos 0  4- cos 50  =  cos 30.  22.   sin 4  0  —  sin 2  0  =  cos 3  0. 

20.  sin  0  4-  sin  5  0  =  sin  3  0.  23.    cos  7  0  —  cos  0  =  —  sin  4 

21.  sin  3  6  +  sin  7  0  =  sin  5  0. 

*The  difference  is  taken,  first  angle  minus  the  second. 


I .  OW^& 


VIII.  §  71]       TRIGONOMETRIC  RELATIONS  103 

MISCELLANEOUS  EXERCISES 

1.  Reduce  to  radians  65°,  -  135°,  -  300°,  20°. 

2.  Reduce  to  degrees  7r,  3  ?r,  —  2  w,  4  v  radians. 

3.  Find  sin  (a  —  /3)  and  cos  (a  +  /S)  when  it  is  given  that  a  and  /3  are 
positiye  and  acute  and  tan  a  =  f  and  sec  /3  =  *£. 

4.  Find  tan  (a  +  /S)  and  tan  (a  —  /3)  when  it  is  given  that  tan  a  =  \ 
and  tan  /S  =  |. 

5.  Prove  that  sin  4  a  =  4  sin  a  cos  a  —  8  sin3  a  cos  a. 

6.  Given  sin  0  =  — -y  and  0  in  the  second  quadrant.     Find  sin  2  0 

V5 
cos  2  0,  tan  2  0. 

Prove  the  following  identities  : 
.      7.   sin2«=    2tan"    ■  9.   sec2«         csc2a 


1  +  tan2  a  esc2  a  —  2 

8.    cos2(,=1-tan2^.  10.    tan«=     sin2a 


1  +  tan2  a  1  +  cos  2  a 

s~  11.   sin  (a  +  /S)  cos  /3  —  cos  (a  +  /3)  sin  0  =  sin  a. 
^-£ll.  sin  2  a  +  sin  2  j3  +  sin  2  7  =  4  sin  a  sin  0  sin  7,  if  a  +  /S  +  7  =  180°. 

1  +  tan  -  c*  ,         q. 

cos  a         2  _d  J  <-~    ^ 

'    l-sta«"l_ta„!'  "=   r  T\^P  & 

—  \_  ten  -x.  4   ^_    1 
.  A  *•  %  <u*J~^&.  ^   _   1 

4^»\A      5i_        .2- 


•  *^-c4^rv^    U^ 


6p 


Jf^iilO    -*q 


sp 


OQ 


0Ti  +#f  T# 


^~  (ot  +ft)      ■*    %*~oC(U->($    *6*->©^ 


u. 


« 


£_    ff    ^^tu#^ 

m 

T<.  uv 

ir.o 

m 

Ho' 

5    R 

M* 

f  *    ' 

^  r 
%  * 

- 

To* 

3 

T? 

fi^L^      I 

/^/^C 

TABLES 

FOUR   DECIMAL  PLACES 


106 

[Moving  the  decimal  poin 


Squares  of  Numbers 

t  one  place  in  N  requires  a  corresponding  move  of  two 
places  in  N2] 


u 

N2    0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.0 

.0000 

.0001 

.0004 

.0009 

.0016 

.0025 

.0036 

.0049 

.0064 

.0081 

0.1 
0.2 
0.3 

0.4 
0.5 
0.6 

0.7 
0.8 
0.9 

.0100 
.0400 
.0900 

.1600 
.2500 
.3600 

.4900 
.6400 

.8100 

.0121 
.0441 
.0961 

.1681 
.2601 
.3721 

.5041 
.6561 

.8281 

.0144 
.0484 
.1024 

.1764 
.2704 
.3844 

.5184 
.6724 
.8464 

.0169 
.0529 
.1089 

.1849 
.2809 
.3969 

.5329 
.6889 
.8649 

.0196 
.0576 
.1156 

.1936 
.2916 
.4096 

.5476 
.7056 
.8836 

.0225 
.0625 
.1225 

.2025 
.3025 
.4225 

.5625 
.7225 
.9025 

.0256 
.0676 
.1296 

.2116 
.3136 
.4356 

.5776 
.7396 
.9216 

.0289 
.0729 
.1369 

.2209 
.3249 
.4489 

.5929 
.7569 
.9409 

.0324 

.0784 
.1444 

.2304 
.3364 
.4624 

.6084 
.7744 
.9604 

.0361 
.0841 
.1521 

.2401 
.3481 
.4761 

.6241 
.7921 
.9801 

1.0 

1.000 

1.020 

1.040 

1.061 

1.082 

1.103 

1.124 

1.145 

1.166 

1.188 

1.1 
1.2 
1.3 

1.4 
1.5 
1.6 

1.7 

1.8 
1.9 

1.210 
1.440 
1.690 

1.960 
2.250 
2.560 

2.890 
3.240 
3.610 

1.232 
1.464 
1.716 

1.988 
2.280 
2.592 

2.924 
3.276 
3.648 

1.254 

1.488 
1.742 

2.016 
2.310 
2.624 

2.958 
3.312 
3.686 

1.277 
1.513 
1.769 

2.045 
2.341 
2.657 

2.993 
3.349 
3.725 

1.300 
1.538 
1.796 

2.074 
2.372 
2.690 

3.028 
3.386 
3.764 

1.323 
1.563 
1.823 

2.103 
2.403 
2.723 

3.063 
3.423 
3.803 

1.346 

1.588 
1.850 

2.132 
2.434 
2.756 

3.098 
3.460 
3.842 

1.369 
1.613 

1.877 

2.161 
2.465 
2.789 

3.133 
3.497 
3.881 

1.392 
1.638 
1.904 

2.190 
2.496 

2.822 

3.168 
3.534 
3.920 

1.416 
1.664 
1.932 

2.220 
2.528 
2.856 

3.204 
3.572 
3.960 

2.0 

4.000 

4.040 

4.080 

4.121 

4.162 

4.203 

4.244 

4.285 

4.326 

4.368 

2.1 
2.2 
2.3 

2.4 
2.5 
2.6 

2.7 

2.8 
2.9 

3.0 

3.1 
3.2 
3.3 

3.4 
3.5 
3.6 

3.7 
3.8 
3.9 

4.410 
4.840 
5.290 

5.760 
6.250 
6.760 

7.290 
7.840 
8.410 

4.452 
4.884 
5.336 

5.808 
6.300 
6.812 

7.344 
7.896 
8.468 

4.494 
4.928 
5.382 

5.856 
6.350 
6.864 

7.398 
7.952 
8.526 

4.537 
4.973 
5.429 

5.905 
6.401 
6.917 

7.453 
8.009 

8.585 

4.580 
5.018 
5.476 

5.954 
6.452 
6.970 

7.508 
8.066 
8.644 

4.623 
5.063 
5.523 

6.003 
6.503 
7.023 

7.563 
8.123 
8.703 

4.666 
5.108 
5.570 

6.052 
6.554 
7.076 

7.618 
8.180 
8.762 

4.709 
5.153 
5.617 

6.101 
6.605 
7.129 

7.573 

8.237 
8.821 

4.652 
5.198 
5.664 

6.150 
6.656 

7.182 

7.728 
8.294 
8.880 

4.796 
5.244 
5.712 

6.200 
6.708 
7.236 

7.784 
8.352 
8.940 

9.000 

9.060 

9.120 

9.181 

9.242 

9.303 

9.364 

9.425 

9.486 

9.548 

9.610 
10.24 
10.89 

11.56 
12.25 
12.96 

13.69 
14.44 
15.21 

9.672 
10.30 
10.96 

11.63 
12.32 
13.03 

13.76 
14.52 
15.29 

9.734 
10.39 
11.02 

11.70 
12.39 
13.10 

13.84 
14.59 
15.37 

9.797 
10.43 
11.09 

11.76 
12.46 
13.18 

13.91 
14.70 
15.44 

9.860 
10.50 
11.16 

11.83 
12.53 
13.25 

13.99 
14.75 
15.52 

9.923 
10.56 
11.22 

11.90 
12.60 
13.32 

14.06 
14.82 
15.60 

9.986 
10.63 
11.29 

11.97 
12.67 
13.40 

14.14 
14.90 
15.68 

10.05 
10.69 
11.36 

12.04 
12.74 
13.47 

14.21 
14.98 
15.76 

10.11 
10.76 
11.42 

12.11 
12.82 
13.54 

14.29 
15.05 
15.84 

10.18 
10.82 
11.49 

12.18 
12.89 
13.62 

14.26 
15.13 
15.92 

4.0 

16.00 

16.08 

16.16 

16.24 

16.32 

16.40 

16.48 

16.56 

16.65 

16.73 

4.1 
4.2 
4.3 

4.4 

4.5 
4.6 

4.7 

4.8 
4.9 

16.81 
17.64 
18.49 

19.36 
20.25 
21.16 

22.09 
23.04 
24.01 

16.89 
17.72 
18.58 

19.45 
20.34 
21.25 

22.18 
23.14 
24.11 

16.97 
17.81 
18.66 

19.54 
20.43 
21.34 

22.28 
23.23 
24.21 

17.06 
17.89 
18.65 

19.62 
20.52 
21.44 

22.37 
23.33 
24.30 

17.14 

17.98 

18.84 

19.71 
20.61 
21.53 

22.47 
23.43 
24.40 

17.22 
18.06 
18.92 

19.80 
20.70 
21.62 

22.56 
23.52 
24.50 

17.31 
18.15 
19.01 

19.89 
20.79 
21.72 

22.66 
23.62 
24.60 

17.39 
18.23 
19.10 

19.98 
20.88 
21.81 

22.75 
23.72 
24.70 

17.47 
18.32 
19.18 

20.07 
20.98 
21.90 

22.85 
23.81 
24.80 

17.56 
18.40 
19.27 

20.16 
21.07 
22.00 

22.94 
23.91 
24.90 

5.0 

25.00 

25.10 

25.20 

25.30 

25.40 

25.50 

25.60 

25.70 

25.81 

25.91 

Squares  of  Numbers 


10' 


[Moving  the  decimal  point  one  place  in  N  requires  a  corresponding  move  of  two 
places  in  N2] 


I 

F  o 

,      . 

3 

4 

5 

6 

7 

8 

9 

5.0 

25.00 

25.10  j  25.20 

25.30 

25.40 

25.50 

25.60 

25.70 

25.81 

25.91 

5.1 
5.2 
5.3 

5.4 
5.5 
5.6 

5.7 

5.8 
5.9 

26.01 
27.04 
28.09 

29.16 
30.25 
31.36 

32.49 
33.64 
34.81 

26.11 
27.14 

28.20 

29.27 
30.36 
31.47 

32.60 
33.76 
34.93 

26.21 
27.25 
28.30 

29.38 
30.47 
31.58 

32.72 
33.87 
35.05 

26.32 
27.35 
28.41 

29.48 
30.58 
31.70 

32.83 
33.99 
35.16 

26.42 
27.46 
28.52 

29.59 
30.69 
31.81 

32.95 
34.11 
35.28 

26.52 
27.56 
28.62 

29.70 
30.80 
31.92 

33.06 
34.22 
35.40 

26.63 
27.67 
28.73 

29.81 
30.91 
32.04 

33.18 
34.34 
35.52 

26.73 
27.77 
28.84 

29.92 
31.02 
32.15 

33.29 
34.46 
35.64 

26.83 
27.88 
28.94 

30.03 
31.14 
32.26 

33.41 
34.57 
35.76 

26.94 
27.98 
29.05 

30.14 
31.25 
32.38 

33.52 
34.69 

35.88 

6.0 

36.00 

36.12 

36.24 

36.36 

36.48 

36.60 

36.72 

36.84 

36.97 

37.09 

6.1 
6.2 
6.3 

6.4 
6.5 
6.6 

6.7 
6.8 
6.9 

37.21 
38.44 
39.69 

40.96 
42.25 
43.56 

44.89 
46.24 
47.61 

37.33 
38.56 
39.82 

41.09 
42.38 
43.69 

45.02 
46.38 
47.75 

37.45 
38.69 
39.94 

41.22 
42.51 
43.82 

45.16 
46.51 
47.89 

37.58 
38.81 
40.07 

41.34 
42.64 
43.96 

45.29 
46.65 
48.02 

37.70 
38.94 
40.20 

41.47 
42.77 
44.09 

45.42 
46.79 
48.16 

37.82 
39.06 
40.32 

41.60 
42.90 
44.22 

45.56 
46.92 
48.30 

37.95 
39.19 
40.45 

41.73 
43.03 
44.36 

45.70 
47.06 
48.44 

38.07 
39.31 
40.58 

41.86 
43.16 
44.49 

45.83 
47.20 
48.58 

38.19 
39.44 
40.70 

41.99 
43.30 
44.62 

45.97 
47.33 

48.72 

38.32 
39.56 
40.83 

42.12 
43.43 
44.76 

46.10 
47.47 

48.72 

7.0 

49.00 

49.14 

49.28 

49.42 

49.56 

49.70 

49.84 

49.98 

50.13 

50.27 

7.1 
7.2 
7.3 

7.4 
7.5 
7.6 

7.7 
7.8 
7.9 

50.41 
51.84 
53.29 

54.76 
56.25 
57.76 

59.29 
60.84 
62.41 

50.55 
51.98 
53.44 

54.91 
56.40 
57.91 

59.44 
61.00 
62.57 

50.69 
52.13 
53.58 

55.06 
56.55 
58.06 

59.60 
61.15 

62.73 

50.84 
52.27 
53.73 

55.20 
56.70 

58.22 

59.75 
61.31 
62.88 

50.98 
52.42 
53.88 

55.35 
56.85 
58.37 

59.91 
61.47 
63.04 

51.12 
52.56 
54.02 

55.50 
57.00 
58.52 

60.06 
61.62 
63.20 

51.27 
52.71 
54.17 

55.65 
57.15 
58.68 

60.22 
61.78 
63.36 

51.41 
52.85 
54.32 

55.80 
57.30 
58.83 

60.37 
61.94 
63.52 

51.55 
53.00 
54.46 

55.95 
57.46 
58.98 

60.53 
62.09 
63.68 

51.70 
£3.14 
54.61 

56.10 
57.61 
59.14 

60.68 
62.25 
63.84 

8.0    64.00 

64.16 

64.32 

64.48    64.64 

64.80 

64.96 

65.12 

65.29 

65.45 

8.1 

8.2 
8.3 

8.4 
8.5 
8.6 

8.7 
8.8 
8.9 

65.61 
67.24 
68.89 

70.56 
72.25 
73.96 

75.69 
77.44 
79.21 

65.77 
67.40 
69.06 

70.73 
72.42 
74.13 

75.86 
77.62 
79.39 

65.93 
67.57 
69.22 

70.90 
72.59 
74.30 

76.04 
77.79 
79.57 

66.10 
67.73 
69.39 

71.06 
72.76 
74.48 

76.21 

77.97 
79.74 

66.26 
67.90 
69.56 

71.23 
72.93 
74.65 

76.39 
78.15 
79.92 

66.42 
68.06 
69.72 

71.40 
73.10 

74.82 

76.56 
78.32 
80.10 

66.59 
68.23 
69.89 

71.57 
73.27 
75.00 

76.74 
78.50 
80.28 

66.75 
68.39 
70.06 

71.74 
73.44 
75.17 

76.91 
78.68 
80.46 

66.91 
68.56 
70.22 

71.91 
73.62 
75.34 

77.08 
78.85 
80.64 

67.08 
68.72 
70.39 

72.08 
73.79 
75.52 

77.26 
79.03 
80.82 

9.0 

81.00 

81.18 

81.36 

81.54 

81.72 

81.90 

82.08 

82.26 

82.45 

82.63 

9.1 
9.2 
9.3 

9.4 
9.5 
9.6 

9.7 
9.8 
9.9 

82.81 
84.64 
86.49 

88.36 
90.25 
92.16 

94.09 
96.04 
98.01 

82.99 
84.82 
86.68 

88.55 
90.44 
92.35 

94.28 
96.24 
98.21 

83.17 
85.00 
86.86 

88.74 
90.63 
92.54 

94.48 
96.43 
98.41 

83.36 
85.19 
87.05 

88.92 
90.82 
92.74 

94.67 
96.63 
98.60 

83.54 
85.38 
87.24 

89.11 
91.01 
92.93 

94.87 
96.83 
98.80 

83.72 
85.56 
87.42 

89.30 
91.20 
93.12 

95.06 
97.02 
99.00 

83.91 
85.75 
87.61 

89.49 
91.39 
93.32 

95.26 
97.22 
99.20 

84.09 
85.93 
87.80 

89.68 
91.58 
93.51 

95.45 
97.42 
99.40 

84.27 
86.12 
87.99 

89.87 
91.78 
93.70 

95.65 
97.61 
99.60 

84.46 
86.30 
88.17 

90.06 
91.97 
93.90 

95.84 
97.81 
99.80 

108 


Powers  and  Roots 


Squares  and  Cubes       Square  Roots  and  Cube  Roots 


No. 

Square 

Cube 

Square 
Eoot 

Cube 
Root 

No. 

Square 

Cube 

Square 
Root 

Cube 
Root 

1 

1 

1 

1.000 

1.000 

51 

2,601 

132,651 

7.141 

3.708 

2 

4 

8 

1.414 

1.260 

52 

2,704 

140,608 

7.211 

3.733 

3 

9 

27 

1.732 

1.442 

53 

2,809 

148,877 

7.280 

3.756 

4 

16 

64 

2.000 

1.587 

54 

2,916 

157,464 

7.348 

3.780 

5 

25 

125 

2.236 

1.710 

55 

3,025 

166,375 

7.416 

3.803 

6 

36 

216 

2.449 

1.817 

56 

3,136 

175,616 

7.483 

3.826 

7 

49 

343 

2.646 

1.913 

57 

3,249 

185,193 

7.550 

3.849 

8 

64 

512 

2.828 

2.000 

58 

3,364 

195,112 

7.616 

3.871 

9 

81 

729 

3.000 

2.080 

59 

3,481 

205,379 

7.681 

3.893 

10 

100 

1,000 

3.162 

2.154 

60 

3,600 

216,000 

7.746 

3.915 

11 

121 

1,331 

3.317 

2.224 

61 

3,721 

226,981 

7.810 

3.936 

12 

144 

1,728 

3.464 

•2.289 

62 

3,844 

238,328 

7.874 

3.958 

13 

169 

2,197 

3.606 

2.351 

63 

3,969 

250,047 

7.937 

3.979 

14 

196 

2,744 

3.742 

2.410 

64 

4,09(5 

262,144 

8.000 

4.000 

15 

225 

3,375 

3.873 

2.466 

65 

4,225 

274,625 

8.062 

4.021 

16 

256 

4,096 

4.000 

2.520 

66 

4,356 

287,496 

8.124 

4.041 

17 

289 

4,913 

4.123 

2.571 

67 

4,489 

300,763 

8.185 

4.0(32 

18 

324 

5,832 

4.243 

2.621 

68 

4,624 

314,432 

8.246 

4.082 

19 

361 

6,859 

4.359 

2.668 

69 

4,761 

328,509 

8.307 

4.102 

20 

400 

8,000 

4.472 

2.714 

70 

4,900 

343,000 

8.367 

4.121 

21 

441 

9,261 

4.583 

2.759 

71 

5,041 

357.911 

8.426 

4.141 

22 

484 

10,648 

4.690 

2.802 

72 

5,184 

373,248 

8.485 

4.1(50 

23 

529 

12,167 

4.796 

2.844 

73 

5,329 

389,017 

8.544 

4.179 

24 

576 

13,824 

4.899 

2.884 

74 

5,476 

405,224 

8.602 

4.198 

25 

625 

15,625 

5.000 

2.924 

75 

5,625 

421,875 

8.660 

4.217 

26 

676 

17,576 

5.099 

2.962 

76 

5,776 

438,976 

8.718 

4.236 

27 

729 

19,683 

5.196 

3.000 

77 

5,929 

456,533 

8.775 

4.254 

28 

784 

21,952 

5.292 

3.037 

78 

6,084 

474,552 

8.832 

4.273 

29 

841 

24,389 

5.385 

3.072 

79 

6,241 

493,039 

8.888 

4.291 

30 

900 

27,000 

5.477 

3.107 

80 

6,400 

512,000 

8.944 

4.309 

31 

961 

29,791 

5.568 

3.141 

81 

6,561 

531,441 

9.000 

4.327 

32 

1,024 

32,768 

5.657 

3.175 

82 

6,724 

551,368 

9.055 

4.344 

33 

1,089 

35,937 

5.745 

3.208 

83 

6,889 

571,787 

9.110 

4.362 

34 

1,156 

39,304 

5.831 

3.240 

84 

7,056 

592,704 

9.165 

4.380 

35 

1,225 

42,875 

5.916 

3.271 

85 

7,225 

614,125 

9.220 

4.397 

36 

1,296 

46,656 

6.000 

3.302 

86 

7,396 

636,056 

9.274 

4.414 

37 

1,369 

50,653 

6.083 

3.332 

87 

7,569 

658,503 

9.327 

4.431 

38 

1,444 

54,872 

6.164 

3.362 

88 

7,744 

681,472 

9.381 

4.448 

39 

1,521 

59,319 

6.245 

3.391 

89 

7,921 

704,969 

9.434 

4.465 

40 

1,600 

64,000 

6.325 

3.420 

90 

8,100 

729,000 

9.487 

4.481 

41 

1,681 

68,921 

6.403 

3.448 

91 

8,281 

753,571 

9.539 

4.498 

42 

1,764 

74,088 

6.481 

3.476 

92 

8,464 

778,688 

9.592 

4.514 

43 

1,849 

79,507 

6.557 

3.503 

93 

8,649 

804,357 

9.644 

4.531 

44 

1,936 

85,184 

6.633 

3.530 

94 

8,836 

830,584 

9.695 

4.547 

45 

2,025 

91,125 

6.708 

3.557 

95 

9,025 

857,375 

9.747 

4.563 

46 

2,116 

97,336 

6.782 

3.583 

96 

9,216 

884,736 

9.798 

4.579 

47 

2,209 

103,823 

6.856 

3.609 

97 

9,409 

912,673 

9.849 

4.595 

48 

2,304 

110,592 

6.928 

3.634 

98 

9,604 

941,192 

9.899 

4.610 

49 

2,401 

117,649 

7.000 

3.659 

99 

9,801 

970,299 

9.950 

4.626 

50 

2,500 

125,000 

7.071 

3.684 

100 

10,000 

1,000,000 

10.000 

4.642 

For  a  more  complete  table,  see  The  Macjuillan  Tables,  pp.  94-111. 


Important  Constants 


109 


Certain  Convenient  Values  for  n  =  1  to  n  =  10 


n 

1/n 

Vn 

■y/n 

n\ 

1/nl 

Logio  11 

1 

1.000000 

1.00000 

1.00000 

1 

1.0000000 

0.000000000 

2 

0500000 

1.41421 

1.25992 

2 

0.5000000 

0.301029996 

3 

0.333333 

1.73205 

1.44225 

6 

0.1666667 

0.477121255 

4 

0.250000 

2.00000 

1.58740 

24 

0.0416667 

0.602059991 

5 

0.200000 

2.23607 

1.70998 

120 

0.0083333 

0.698970004 

6 

0.166667 

2.44949 

1.81712 

720 

0.0013889 

0.778151250 

7 

0.142857 

2.64575 

1.91293 

5040 

0.0001984 

0.845098040 

8 

0.125000 

2.82843 

2.00000 

40320 

0.0000248 

0.903089987 

9 

0.111111 

3.00000 

2.08008 

362880 

0.0000028 

0.954242509 

10 

0.100000 

3.16228 

2.15443 

3628800 

0.0000003 

1.000000000 

Logarithms  of  Important  Constants 


71  =■  NUMBER 

Value  of  n 

Log  io  n 

IT 

3.14159265 

0.49714987 

1-4-  7T 

0.31830989 

9.50285013 

7r2 

9.86960440 

0.99429975 

VtF 

1.77245385 

0.24857494 

e  =  Napierian  Base 

2.71828183 

0.43429448 

M=  logw  e 

0.43429448 

9.63778431 

l-5-if=loge10 

2.30258509 

0.36221569 

180  -7-  7r  =  degrees  in  1  radian 

57.2957795 

1.75812262 

7r  -r- 180  =  radians  in  1° 

0.01745329 

8.24187738 

ir  -4- 10800  =  radians  in  1' 

0.0002908882 

6.46372613 

t  -7-  648000  =  radians  in  1" 

0.000004848136811095 

4.68557487 

sin  1" 

0.000004848136811076 

4.68557487 

tan  1" 

0.000004848136811152 

4.68557487 

centimeters  in  1  ft. 

30.480 

1.4840158 

feet  in  1  cm. 

0.032808 

8.5159842 

inches  in  1  m. 

39.37  (exact  legal  value) 

1.5951654 

pounds  in  1  kg. 

2.20462 

0.3433340 

kilograms  in  1  lb. 

0.453593 

9.6566660 

g  (average  value) 

32.16  ft./sec./sec. 

1.5073 

=  981  cm./sec/sec 

2.9916690 

weight  of  1  cu.  ft.  of  water 

62.425  lb.  (max.  density) 

1.7953586 

weight  of  1  cu.  ft.  of  air 

0.0807  lb.  (at  32°  F.) 

8.907 

cu.  in.  in  1  (U.  S.)  gallon 

231  (exact  legal  value) 

2.3636120 

ft.  lb.  per  sec.  in  1  H.  P. 

550.  (exact  legal  value) 

2.7403627 

kg.  m.  per  sec.  in  1  H.  P. 

76.0404 

1.8810445 

watts  in  1  H.  P. 

745.957 

2.8727135 

11C 

1 

Fo 

ur  ] 

*lac 

e  L( 

)gar 

ithr 

US 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12  3 

4  5  6 

7  8  9 

10 

0000 

0043 

0080 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

4  8  12 

17  21  25 

29  33  37 

11 

12 
13 

14 
15 
16 

17 

18 
19 

0414 
0792 
1139 

1461 
1761 
2041 

2304 

2553 
2788 

0453 

0828 
1173 

1492 

1790 
2068 

2330 
2577 
2810 

0492 
0864 
1206 

1523 

1818 
2095 

2355 
2601 
2833 

0531 
0899 
1239 

1553 
1847 
2122 

2380 
2625 
2856 

0569 
0934 
1271 

1584 
1875 
2148 

2405 
2648 

2878 

0607 
0969 
*1303 

1614 
1903 
2175 

2430 
2672 
2900 

0645 
1004 
1335 

1644 
1931 
2201 

2455 

2695 
2923 

0682 
1038 
1367 

1673 
1959 

2227 

2480 
2718 
2945 

0719 
1072 
1399 

1703 
1987 
2253 

2504 
2742 
2967 

0755 
1106 
1430 

1732 
2014 
2279 

2529 
2765 
2989 

4  8  11 
3  7  10 
3  6  10 

3  6  9 
3  6  8 
3  5  8 

2  5  7 
2  5  7 
2  4  7 

15  If-  23 
14  17  21 
13  16  19 

12  15  18 
11  14  17 
11  13  16 

10  12  15 
9  12  14 
9  11  13 

26  30  34 
24  28  31 
23  26  29 

21  24  27 
20  22  25 
18  21  24 

17  20  22 
16  19  21 

16  18  20 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

2  4  6 

8  1113 

15  17  19 

21 

22 
23 

24 
25 

26 

27 

1 

3222 
3424 
3617 

3802 
3979 
4150 

4314 
4472 
4624 

3243 
3444 
3636 

3820 
3997 
4166 

4330 

4487 
4639 

3263 
3464 
3655 

3838 
4014 
4183 

4346 

4502 
4654 

3284 
3483 
3674 

3856 
4031 
4200 

4362 
4518 
4669 

3304 
3502 
3692 

3874 
4048 
4216 

4378 
4533 
4683 

3324 
3522 
3711 

3892 
4065 
4232 

4393 

4548 
4698 

3345 
3541 
3729 

3909 
4082 
.4249 

4409 
4564 
4713 

3365 
3560 
3747 

3927 
4099 
4265 

4425 
4579 

4728 

3385 
3579 
3766 

3945 

4110 
4281 

4440 
4594 
4742 

3404 
3598 
3784 

3962 
4133 

4298 

4456 
4609 
4757 

2  4  6 
2  4  6 
2  4  6 

2  4  5 
2  4  5 
2  3  5 

2  3  5 
2  3  5 
13  4 

8  10  12 
8  10  12 
7  9  11 

7  9  11 
7  9  10 

7  8  10 

6  8  9 
6  8  9 
6  7  9 

14  16  18 
14  16  17 
13  15  17 

12  14  16 
12  14  16 
11  13  15 

11  12  14 
11  12  14 
10  12  13 

30 

31 
32 
33 

34 
35 

36 

37 
38 
39 

4771 

4786 

4928 
5065 
5198 

5328 
5453 
5575 

5694 

5809 
5922 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

13  4 

6  7  9 

10  11 13 

4914 
5051 
5185 

5315 
5441 
5563 

5682 
5798 
5911 

4942 
5079 
5211 

5340 
5405 
5587 

5705 
5821 
5933 

4955 
5092 
5224 

5353 
5478 
5599 

5717 

5832 
5944 

4969 
5105 
5237 

5366 
5490 
5611 

5729 
5843 
5955 

4983 
5119 
5250 

5378 
5502 
5623 

5740 

5855 
5966 

4997 
5132 
5263 

5391 
5514 
5635 

5752 

5866 
5977 

5011 
5145 
5276 

5403 

5527 
5647 

5763 
5877 
5988 

5024 
5159 
5289 

5416 

5539 
5658 

5775 

5888 
5999 

5038 
5172 
5302 

5428 
5551 
5670 

5786 
5899 
6010 

13  4 
13  4 
13  4 

12  4 
12  4 
12  4 

12  4 
1  2  3 
1  2  3 

5  7  8 
5  7  8 
5  7  8 

5  6  8 
5  6  7 
5  6  7 

5  6  7 
5  6  7 
4  5  7 

10  11  12 
91112 
9  1112 

9  10  11 
9  10  11 
8  1011 

8  911 
8  9  10 
8  910 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

12  3 

4  5  6 

8  9  10 

41 
42 
43 

44 
45 

46 

47 
48 
49 

6128 
6232 
6335 

6435 
6532 
6628 

6721 
6812 
6902 

6138 
6213 
6345 

6444 
6542 
6637 

6730 
6821 
6911 

6149 
6253 
6355 

6454 
6551 
6646 

6739 
6830 
6920 

6160 
6263 
6365 

6464 
6561 
6656 

6749 
6839 
6928 

6170 
6274 
6375 

6474 
6571 
6665 

6758 
6848 
6937 

6180 
6284 
6385 

6484 
6580 
6675 

6767 
6857 
6946 

6191 
6294 
6395 

6493 
6590 
6684 

6776 
6866 
6955 

6201 
6304 
6405 

6503 
6599 
6693 

6785 
6875 
6964 

6212 
6314 
6415 

6513 
6609 
6702 

6794 
6884 
6972 

6222 
6325 
6425 

6522 
6618 
6712 

6803 
6893 
6981 

12  3 
12  3 
12  3 

12  3 
12  3 
12  3 

12  3 

12  3 
12  3 

4  5  6 
4  5  6 
4  5  6 

4  5  6 
4  5  6 
4  5  6 

4  5  6 
4  5  6 
4  4  5 

7  8  9 
7  8  9 

7  8  9 

7  8  9 
7  8  9 
7  7  8 

7  7  8 
7  7  8 
6  7  8 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

12  3 

3  4  5 

6  7  8 

51 
52 
53 

54 

7076 
7160 
7243 

7324 

7084 
7168 
7251 

7332 

7093 

7177 
7259 

7340 

7101 
7185 
7267 

7348 

7110 
7193 

7275 

7356 

7118 
7202 
7284 

7364 

7126 
7210 
7292 

7372 

7135 
7218 
7300 

7380 

7143 

7226 
7308 

7388 

7152 
7235 
7316 

7396 

12  3 
12  3 
12  2 

12  2 

3  4  5 
3  4  5 
3  4  5 

3  4  5 

6  7  8 
6  7  7 
6  6  7 

6  6  7 

I 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12  2 

4  5  6 

7  8  9 

The  proportional  parts  are  stated  in  fall  for  every  tenth  at  the  right-hand  side. 
The  logarithm  of  any  number  of  four  significant  figures  can  be  read  directly  by  add- 


Four  Place  Logarithms 

111 

If 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12   3 

4    5    6 

7    8    9 

55 

56 

57 
58 
59 

7404 
7482 

7559 
7634 
7709 

7412 
7490 

7566 
7642 
7716 

7419 
7497 

7574 
7649 
7723 

7427 
7505 

7582 
7657 
7731 

7435 
7513 

7589 
7664 

7738 

7443 
7520 

7597 
7672 
7745 

7451 

7528 

7604 
7679 

7752 

7459 
7536 

7612 
7686 
7760 

7466 
7543 

7619 
7694 
7767 

7474 
7551 

7627 
7704 
7774 

12    2 
12    2 

1    1    2 
1    1    2 
112 

3    4    5 
34   5 

3   4    5 
3   4   4 
3   4   4 

5  6  7 
5   6   7 

5  6  7 
5  6  7 
5    6   7 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

112 

3   4   4 

5    6    6 

61 
62 
63 

64 
65 

66 

67 
68 
69 

7853 
7924 
7993 

8062 
8129 
8195 

8261 
8325 
8388 

7860 
7931 
8000 

8069 
8136 
8202 

8267 
8331 
8398 

7868 

e 

8075 
8142 
8209 

8274 
83:58 
8401 

7875 
7945 
8014 

8082 
8149 
8215 

8280 
8344 
8407 

7882 
7952 
8021 

8089 
8156 
8222 

8287 
8351 
8414 

7889 
7959 
8028 

8096 
8162 
8228 

8293 
8357 
8420 

7896 
7966 
8035 

8102 
8169 
8235 

8299 
8363 
8426 

7903  7910  7917 
7973  7980  7987 
8041  8048  8055 

8109  8116  8122 
8176  8182  8189 
8241  8248  8254 

8306  8312  8319 
8370  8376  8382 
8432  8439|  8445 

112 
1    1    2 
112 

112 
112 
112 

112 
112 
112 

3    3   4 
3    3   4 
3   3   4 

3    3   4 
3   3   4 
3   3   4 

3    3   4 
3    3   4 
3    3   4 

5  6  6 
5  5  6 
5    5   6 

5  5  6 
5  5  6 
5    5   6 

5  5  6 
4  5  6 
4    5    6 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500  8506 

112 

3   3    4 

4   5   6 

71 
72 
73 

74 
75 

76 

77 
78 
79 

8513 
8573 
8633 

8692 
8751 
8808 

8865 
8921 
8976 

8519 
8579 
8639 

8698 
8756 
8814 

8871 
8927 

8982 

8525 
8585 
8645 

8704 
8762 
8820 

8876 

8932 
8987 

8531 
8591 
8651 

8710 
8768 
8825 

8882 
8938 
8993 

8537 
8597 
8657 

8716 

8774 
8831 

8887 
8943 
8998 

8543 
8603 
8663 

8722 
8779 
8837 

8893 
8949 
9004 

8549 

8609 
8669 

8727 
8785 
8842 

8899 
8954 
9009 

8555 
8615 
8675 

87&3 
8791 

8848 

8904 
8960 
9015 

8561 
8621 
8681 

8739 
8797 
8854 

8910 
8965 
9020 

8567 
8627 
8686 

8745 
8802 
8859 

8915 
8971 
9025 

112 
112 
112 

112 
112 
112 

1    1    2 
112 
112 

3    3   4 
3    3   4 
2    3   4 

2    3   4 
2    3    3 
2    3   3 

2    3    3 
2    3   3 
2    3   3 

4  5  6 
4  5  6 
4    5    5 

4  5  5 
4  5  5 
4    4   5 

4  4  5 
4  4  5 
4    4    5 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069|  9074 

9079 

1    1    2 

2    3    3 

4    4    5 

81 
82 
83 

84 
85 

86, 

87 
88 
89 

90&5 
9138 
9191 

9243 
9294 
9345 

9395 
9445 
9494 

9090 
9143 
9196 

9248 
9299 
9350 

9400 
9450 
9499 

9096 
9149 
9201 

9253 
9304 
9355 

9405 
9455 
9504 

9101 
9154 
9206 

9258 
9309 
9360 

9410 
9460 
9509 

910(5 
9159 
9212 

9263 
9315 
9365 

9415 
9465 
9513 

9112 
9165 
9217 

9269 
9320 
9370 

9420 
9469 
9518 

9117 
9170 
9222 

9274 
9325 
9375 

9425 
9474 
9523 

9122 
9175 
9227 

9279 
9330 
9380 

9430 
9479 
9528 

9128 
9180 
9232 

9284 
93,35 
9385 

9435 
9484 
9533 

9133 
9186 
9238 

9289 
9340 
9390 

9440 

9489 
9538 

112 
112 
112 

112 
112 
112 

112 
Oil 
0    1    1 

2    3    3 
2   3    3 
2    3   3 

2    3    3 
2    3   3 
2   3   3 

2    3   3 
2   2    3 

2    2    3 

4   4   5 

4  4  5 
4   4   5 

4  4  5 
4  4  5 
4   4   5 

4  4  5 
3  4  4 
3    4    4 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

958(5 

Oil 

2    2    3 

3   4    4 

91 
92 
93 

94 
95 
96 

97 
98 
99 

9590 
9638 
9685 

9731 
9777 
9823 

9868 
9912 
9956 

9595 
9643 
9689 

9736 
9782 
9827 

9872 
9917 
9961 

9600 
9647 
9694 

9741 

9786 
9832 

9877 
9921 
9965 

9605 
9652 
9699 

9745 
9791 
9836 

9881 
9926 
9969 

9609 
9657 
9703 

9750 
9795 
9841 

9886 
993(1 
9974 

9614 

9661 
9708 

9754 
9800 
9845 

9890 
9934 
9978 

9619 
9666 
9713 

9759 

9805 
9850 

9894 
9939 
9983 

9624 
9671 
9717 

9763 
9809 
9854 

9899 
9943 
9987 

9628 
9675 
9722 

9768 
9814 
9859 

9903 
9948 
9991 

9633 
9680 
9727 

9773 

9818 
9863 

9908 
9952 
9996 

0    1    1 
Oil 
Oil 

Oil 
Oil 
Oil 

0    1    1 
Oil 
0    1    1 

2    2    3 
2   2    3 

2    2    3 

2    2    3 
2   2   3 

2    2    3 

2    2    3 
2    2    3 

2    2    3 

3  4  4 
3   4   4 

3    4    4 

3  4  4 
3  4  4 
3   4    4 

3  4  4 
3  3  4 
3    3   4 

N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

12    3 

4    5    6 

7    8    9 

ing  the  proportional  part  corresponding  to  the  fourth  figure  to  the  tabular  number 
corresponding  to  the  first  three  figures.    There  may  be  an  error  of  1  in  the  last  place. 


112 


Four  Place  Trigonometric  Functions 


[Characteristics  of  Logarithms  omitted  — 

determine  by  the  usual  rule  from  the  value] 

Radians 

Degbees 

Sine 

Tangent 

Cotangent 

Cosine 

Value 

Log10 

Value  Log10 

Value 

Logio 

Value 

Log10 

.0000 
.0029 

0°00' 

10 

.0000 
.0029 

.0000  

.0029  .4637 

1.0000 

.0000 

90°  00' 

-50 

1.5708 
1.5679 

.4637 

343.77 

.5363 

i!oooo 

!oooo 

.0058 

20 

.0058 

.7648 

.0058  .7648 

171.89 

.2352 

1.0000 

.0000 

40 

1.5650 

.0087 

30 

.0087 

.9408 

.0087  .9409 

114.59 

.0591 

1.0000 

.0000 

3Q 

1.5621 

.0116 

40 

.0116 

.0658 

.0116  .0658 

85.940 

.9342 

.9999 

.0000 

20 

1.5592 

.0145 

50 

.0145 

.1627 

.0145  .1627 

68.750 

.8373 

.9999 

.0000 

10 

1.5563 

.0175 

1°00' 

.0175 

.2419 

.0175  .2419 

57.290 

.7581 

.9998 

.9999 

89°  00' 

1.5533 

.0204 

10 

.0204 

.3088 

.0204  .3089 

49.104 

.6911 

.9998 

.9999 

50 

1.5504 

.0233 

20 

.0233 

.3668 

.0233  .3669 

42.964 

.6331 

.9997 

.9999 

40 

1.5475 

.0262 

30 

.0262 

.4179 

.0262  .4181 

38.188 

.5819 

.9997 

.9999 

30 

1.5446 

.0291 

40 

.0291 

.4637 

.0291  .4638 

34.368 

.5362 

.9996 

.9998 

20 

1.5417 

.0320 

50 

.0320 

.5050 

.0320  .5053 

31.242 

.4947 

.9995 

.9998 

10 

1.5388 

.0349 

2°  00' 

.0349 

.5428 

.0349  .5431 

28.636 

.4569 

.9994 

.9997 

88°  00' 

1.5359 

.0378 

10 

.0378 

.5776 

.0378  .5779 

26.432 

.4221 

.9993 

.9997 

50 

1.5330 

.0407 

20 

.0407 

.6097 

.0407  .6101 

24.542 

.3899 

.9992 

.9996 

40 

1.5301 

.0436 

30 

.0436 

.6397 

.0437  .6401 

22.904 

.3599 

.9990 

.9996 

30 

1.5272 

.0465 

40 

.0465 

.6677 

.0466  .6682 

21.470 

.3318 

.9989 

.9995 

20 

1.5243 

.0495 

50 

.0494 

.6940 

.0495  .6945 

20.20(3 

.3055 

.9988 

.9995 

10 

1.5213 

.0524 

3°  00' 

.0523 

.7188 

.0524  .7194 

19.081 

.2806 

.9986 

.9994 

87°  00' 

1.5184 

.0553 

10 

.0552 

.7423 

.0553  .7429 

18.075 

.2571 

.9985 

.9993 

50 

1.5155 

.0582 

20 

.0581 

.7645 

.0582  .7652 

17.169 

.2348 

.9983 

.9993 

40 

1.5126 

.0611 

30 

.0610 

.7857 

.0612  .7865 

16.350 

.2135 

.9981 

.9992 

30 

1.5097 

.0640 

40 

.0640 

.8059 

.0641  .8067 

15.605 

.1933 

.9980 

.9991 

20 

1.5068 

.0669 

50 

.0669 

.8251 

.0670  .8261 

14.924 

.1739 

.9978 

.9990 

10 

1.5039 

.0698 

4°  00' 

.0698 

.8436 

.0699  .8446 

14.301 

.1554 

.9976 

.9989 

86°  00' 

1.5010 

.0727 

10 

.0727 

.8613 

.0729  .8624 

13.727 

.1376 

.9974 

.9989 

50 

1.4981 

.0756 

20 

.0756 

.8783 

.0758  .8795 

13.197 

.1205 

.9971 

.9988 

40 

1.4952 

.0785 

30 

.0785 

.8946 

.0787  .8960 

12.706 

.1040 

.9969 

.9987 

30 

1.4923 

.0814 

40 

.0814 

.9104 

.0816  .9118 

12.251 

.0882 

.9967 

.9986 

20 

1.4893 

.0844 

50 

.0843 

.9256 

.0846  .9272 

11.826 

.0728 

.9964 

.9985 

10 

1.4864 

.0873 

5°  00' 

.0872 

.9403 

.0875  .9420 

11.430 

.0580 

.9962 

.9983 

85° 00' 

1.4835 

.0902 

10 

.0901 

.9545 

.0904  .9563 

11.059 

.0437 

.9959 

.9982 

50 

1.4806 

.0931 

20 

.0929 

.9682 

.0934  .9701 

10.712 

.0299 

.9957 

.9981 

40 

1.4777 

.0960 

30 

.0958 

.9816 

.0963  .9836 

10.385 

.0164 

.9954 

.9980 

30 

1.4748 

.0989 

40 

.0987 

.9945 

.0992  .9966 

10.078 

.0034 

.9951 

.9979 

20 

1.4719 

.1018 

50 

.1016 

.0070 

.1022  .0093 

9.7882 

.9907 

.9948 

.9977 

10 

1.4690 

.1047 

6°  00' 

.1045 

.0192 

.1051  .0216 

9.5144 

.9784 

.9945 

.9976 

84°  00' 

1.4661 

.1076 

10 

.1074 

.0311 

.1080  .0336 

9.2553 

.9664 

.9942 

.9975 

50 

1.4632 

.1105 

20 

.1103 

.0426 

.1110  .0453 

9.0098 

.9547 

.9939 

.9973 

40 

1.4603 

.1134 

30 

.1132 

.0539 

.1139  .0567 

8.7769 

.9433 

.9936 

.9972 

30 

1.4573 

.1164 

40 

.1161 

.0648 

.1169  .0678 

8.5555 

.9322 

.9932 

.9971 

20 

1.4544 

.1193 

50 

.1190 

.0755 

.1198  .0786 

8.3450 

.9214 

.9929 

.9969 

10 

1.4515 

.1222 

7°  00' 

.1219 

.0859 

.1228  .0891 

8.1443 

.9109 

.9925 

.9968 

83°  00' 

1.4486 

.1251 

10 

.1248 

.0961 

.1257  .0995 

7.9530 

.9005 

.9922 

.9966 

/50 

1.4457 

.1280 

20 

.1276 

.1060 

.1287  .1096 

7.7704 

.8904 

.9918 

.9964 

r40 

*30 

1.4428 

.1309 

30 

.1305 

.1157 

.1317  .1194 

7.5958 

.8806 

.9914 

.9963 

1.4399 

.1338 

40 

.1334 

.1252 

.1346  .1291 

7.4287 

.8709 

.9911 

.9961 

20 

1.4370 

.1367 

50 

.1363 

.1345 

.1376  .1385 

7.2687 

.8615 

.9907 

.9959 

10 

1.4341 

.1396 

8°  00' 

.1392 

.1436 

.1405  .1478 

7.1154 

.8522 

.9903 

.9958 

82°  00' 

1.4312 

.1425 

10 

.1421 

.1525 

.1435  .1569 

6.9682 

.8431 

.9899 

.9956 

50 

1.4283 

.1454 

20 

.1449 

.1612 

.1465  .1658 

6.8269 

.8342 

.9894 

.9954 

40 

1.4254 

.1484 

30 

.1478 

.1697 

.1495  .1745 

6.6912 

.8255 

.9890 

.9952 

30 

1.4224 

.1513 

40 

.1507 

.1781 

.1524  .1831 

6.5606 

.8169 

.988(5 

.9950 

20 

1.4195 

.1542 

50 

.1536 

.1863 

.1554  .1915 

6.4348 

.8085 

.9881 

.9948 

10 

1.4166 

.1571 

9°  00' 

.1564 

.1943 

.1584  .1997 

6.3138 

.8003 

.9877 

.9946 

81°  00' 

1.4137 

Value 

Log10 

Value  Lojr10 

Value 

Log10 

Value 

Log10 

Degrees 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

Four  Place  Trigonometric  Functions 


113 


[Characteristics  of  Logarithms  omitted  — 

ietermine  by  the  usual  rule  from  the  value] 

Radians 

Degbees 

Sine 

Tangent 

Cotangent 

Cosine 

Value 

Log10 

Value 

Logio 

Value   Log10 

Value 

L«g10 

.1571 

9°  00' 

.1564 

.1943 

.1584 

.1997 

6.3138  .8003 

.9877 

.9946 

81°  00' 

1.4137 

.1600 

10 

.1593 

.2022 

.1614 

.2078 

6.1970  .7922 

.9872 

.9944 

50 

1.4108 

.1629 

20 

.1622 

.2100 

.1644 

.2158 

6.0844  .7842 

.9868 

.9942 

40 

1.4079 

.1658 

30 

.1650 

.2176 

.1673 

.2236 

5.9758  .7764 

L9868 

'.9858 

.9940 

30 

1.4050 

.1687 

40 

.1679 

.2251 

.1703 

.2313 

5.8708  .7687 

.9938 

20 

1.4021 

.1716 

50 

.1708 

.2324 

.1733 

.2389 

5.7694  .7611 

.9853 

.9936 

10 

1.3992 

.1745 

10°  00 

.1736 

.2397 

.1763 

.2463 

5.6713  .7537 

.9848 

.9934 

80°  00' 

1.3963 

.1774 

10 

.1765 

.2468 

.1793 

.2536 

5.5764  .7464 

.9843 

.9931 

50 

1.3934 

.1804 

20 

.1794 

.2538 

.1823 

.2609 

5.4845  .7391 

.9838 

.9929 

40 

1.3904 

.1833 

30 

.1822 

.2606 

.1853 

.2680 

5.3955  .7320 

.9833 

.9927 

30 

1.3875 

.1862 

40 

.1851 

.2674 

.1883 

.2750 

5.3093  .7250 

.9827 

.9924 

20 

1.3846 

.1891 

50 

.1880 

.2740 

.1914 

.2819 

5.2257  .7181 

.9822 

.9922 

10 

1.3817 

.1920 

11°00' 

.1908 

.2806 

.1944 

.2887 

,5.1446  .7113 

.9816 

.9919 

79°  00 

1.3788 

.1949 

10 

.1937 

.2870 

:1974 

.2953 

5.0658  .7047 

.9811 

.9917 

50 

1.3759 

.1978 

20 

.1965 

.2934 

.2004 

.3020 

4.9894  .6980 

.9805 

.9914 

40 

1.3730 

.2007 

30 

.1994 

.2997 

.2035 

.3085 

4.9152  .6915 

.9799 

.9912 

30 

1.3701 

.2036 

4D 

.2022 

.3058 

.2065 

.3149 

4.8430  .6851 

.9793 

.9909 

20 

1.3672 

.2065 

50 

.2051 

.3119 

.2095 

.3212 

4.7729  .6788 

.9787 

.9907 

10 

1.3643 

.2094 

12°  00' 

.2079 

.3179 

.2126 

.3275 

4.7046  .6725 

.9781 

.9904 

78°  00' 

L3614 

.2123 

10 

.2108 

.3238 

.2156 

.3336 

4.6382  .6664 

.9775 

.9901 

50 

1.3584 

.2153 

20 

.2136 

.3296 

.2186 

.3397 

4.5736  .6603 

.9769 

.9899 

40 

1.3555 

.2182 

30 

.2164 

.3353 

.2217 

.3458 

•4.5107  .6542 

.9763 

.9896 

30 

1.3526 

1  .2211 

40 

.2193 

.3410 

.2247 

.3517 

4.4494  .6483 

.9757 

.9893 

20 

1.3497 

.2240 

50 

.2221 

.3466 

.2278 

.3576 

4.3897  .6424 

.9750 

.9890 

10 

1.3468 

.2269 

13°  00' 

.2250 

.3521 

.2309 

.3634 

4.3315  .6366 

.9744 

.9887 

77°  00' 

1.3439 

.2298 

10 

.2278 

.3575 

.2339 

.3691 

4.2747  .6309 

.9737 

.9884 

50 

1.3410 

.2327 

20 

.2306 

.3629 

.2370 

.3748 

4.2193  .6252 

.9730 

.9881 

40 

1.3381 

.2356 

30 

.2334 

.3682 

.2401 

.3804 

4.1653  .6196 

.9724 

.9878 

30 

1.3352 

.2385 

40 

.2363 

.3734 

.2432 

.3859 

4.1126  .6141 

.9717 

.9875 

20 

1.3323 

.2414 

50 

.2391 

.3786 

.2462 

.3914 

4.0611  .6086 

.9710 

.9872 

10 

1.3294 

.2443 

14°  00' 

.2419 

.3837 

.2493 

.3968 

4.0108  .6032 

.9703 

.9869 

76°  00' 

1.3265 

.2473 

10 

.2447 

.3887 

.2524 

.4021 

3.9617  .5979 

.9696 

.9866 

50 

1.3235 

.2502 

20 

.2476 

.3937 

.2555 

.4074 

3.9136  .5926 

.9689 

.9863 

40 

1.3206 

.2531 

30 

.2504 

.3986 

.2586 

.4127 

3.8667  .5873 

.9681 

.9859 

30 

1.3177 

.2560 

40 

.2532 

.4035 

.2617 

.4178 

3.8208  .5822 

.9674 

.9856 

20 

1.3148 

.2589 

50 

.2560 

.4083 

.2648 

.4230 

3.7760  .5770 

.9667 

.9853 

10 

1.3119 

.2618 

15°00' 

.2588 

.4130 

.2679 

.4281 

3.7321  .5719 

.9659 

.9849 

75°  00' 

1.3090 

.2647 

10 

.2616 

.4177 

.2711 

.4331 

3.6891  .5669 

.9652 

.9846 

50 

1.3061 

.2676 

20 

.2644 

.4223 

.2742 

.4381 

3.6470  .5619 

.9644 

.9843 

40 

1.3032 

.2705 

30 

.2672 

.4269 

.2773 

.4430 

3.6059  .5570 

.9636 

.9839 

30 

1.3003 

.2734 

40 

.2700 

.4314 

.2805 

.4479 

3.5656  .5521 

.9628 

.9836 

20 

1.2974 

.2763 

50 

.2728 

.4359 

.2836 

.4527 

3.5261  .5473 

.9621 

.9832 

10 

1.2945 

.2793 

16°  00' 

.2756 

.4403 

.2867 

.4575 

3.4874  .5425 

.9613 

.9828 

74°  00' 

1.2915 

.2822 

10 

.2784 

.4447 

.2899 

.4622 

3.4495  .5378 

.9605 

.9825 

50 

1.2886 

.2851 

20 

.2812 

.4491 

.2931 

.4669 

3.4124  .5331 

.9596 

.9821 

40 

1.2857 

.2880 

30 

.2840 

.4533 

.2962 

.4716 

3.3759  .5284 

.9588 

.9817 

30 

1.2828 

.2909 

40 

.2868 

.4576 

.2994 

.4762 

3.3402  .5238 

.9580 

.9814 

20 

1.2799 

.2938 

50 

.2896 

.4618 

.3026 

.4808 

3.3052  .5192 

.9572 

.9810 

10 

1.2770 

.2967 

17°  00' 

.2924 

.4659 

.3057 

.4853 

3.2709  .5147 

.9563 

.9806 

73°  00' 

1.2741 

.2996 

10 

.2952 

.4700 

.3089 

.4898 

3.2371  .5102 

.9555 

.9802 

50 

1.2712 

.3025 

20 

.2979 

.4741 

.3121 

.4943 

3.2041  .5057 

.9546 

.9798 

40 

1.2683 

.3054 

30 

.3007 

.4781 

.3153 

.4987 

3.1716  .5013 

.9537 

.9794 

30 

1.2654 

.3083 

40 

.3035 

.4821 

.3185 

.5031 

3.1397  .4969 

.9528 

.9790 

20 

1.2625 

.3113 

50 

.3062 

.4861 

.3217 

.5075 

3.1084  .4925 

.9520 

.9786 

10 

1.2595 

.3142 

18°  00' 

.3090 

.4900 

.3249 

.5118 

3.0777  .4882 

.9511 

.9782 

72°  00' 

1.2566 

Value 

Logio 

Value 

Log10 

Value   Log10 

Value 

Log10 

Degrees 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

114 


Four  Place  Trigonometric  Functions 


[Characteristics  of  Logarith 

ms  omitted  — 

determine  by  the  usual  rule  from  the  value]' 

Radians 

Degrees 

Sine 

Tangent 

Cotangent 

Cosine 

Value 

L°g10 

Value 

Logx 

Value 

Log10 

Value  Log10 

.3142 

18°  00' 

.3090 

.4900 

.3249 

.5118 

3.0777 

.4882 

.9511  .9782 

72°  00' 

1.2566 

.3171 

10 

.3118 

.4939 

.3281 

.5161 

3.0475 

.4839 

.9502  .9778 

50 

1 .2537 

.3200 

20 

.3145 

.4977 

.3314 

.5203 

3.0178 

.4797 

.9492  .9774 

40 

1.2508 

.3229 

30 

.3173 

.5015 

.3346 

.5245 

2.9887 

.4755 

.9483  .9770 

30 

1.2479 

.3258 

40 

.3201 

.5052 

.3378 

.5287 

2.9600 

.4713 

.9474  .9765 

20 

1.2450 

.3287 

50 

.3228 

.5090 

.3411 

.5329 

2.9319 

.4671 

.9465  .9761 

10 

1.2421 

.3316 

19°  00' 

.3256 

.5126 

.3443 

.5370 

2.9042 

.4630 

.9455  .9757 

71° 00' 

1.2392 

.3345 

10 

.3283 

.5163 

.3476 

.5411 

2.8770 

.4589 

.9446  .9752 

50 

1.2363 

.3374 

20 

.3311 

.5199 

.3508 

.5451 

2.8502 

.4549 

.9436  .9748 

40 

1.2334 

.3403 

30 

.3338 

.5235 

.3541 

.5491 

2.8239 

.4509 

.9426  .9743 

30 

1.2305 

.3432 

40 

.3365 

.5270 

.3574 

.5531 

2.7980 

.4469 

.9417  .9739 

20 

1.2275 

.3462 

50 

.3393 

.5306 

.3607 

.5571 

2.7725 

.4429 

.9407  .9734 

10 

1.2246 

.3491 

20°  00' 

.3420 

.5341 

.3640 

.5611 

2.7475 

.4389 

.9397  !9730 

70°  00' 

1.2217 

.3520 

10 

.3448 

.5375 

.3073 

.5650 

2.7228 

.4350 

.9387  .9725 

50 

1.2188 

.3549 

20 

.3475 

.5409 

.3706 

.5689 

2.6985 

.4311 

.9377  .9721 

40 

1.2159 

.3578 

30 

.3502 

.5443 

.3739 

.5727 

2.6746 

.4273 

.9367  .9716 

30 

1.2130 

.3607 

40 

.3529 

.5477 

.3772 

.5766 

2.6511 

.4234 

.9356  .9711 

20 

1.2101 

.3636 

50 

.3557 

.5510 

.3805 

.5804 

2.6279 

.4196 

.9346  .9706 

10 

1.2072 

.3665 

21°  00' 

.3584 

.5543 

.3839 

.5842 

2.6051 

.4158 

.9336  ,9702 

69°  00' 

1.2043 

.3694 

10 

.3611 

.5576 

.3872 

.5879 

2.5826 

.4121 

.9325  .9697 

50 

1.2014 

.3723 

20 

.3638 

.5609 

.3906 

.5917 

2.5605 

.4083 

.9315  .9692 

40 

1.1985 

.3752 

30 

.3665 

.5641 

.3939 

.5954 

2.5386 

.4046 

.9304  .9687 

30 

1.1956 

.3782 

40 

.3692 

.5673 

.3973 

.5991 

2.5172 

.4009 

.9293  .9682 

20 

1.1926 

.3811 

50 

.3719 

.5704 

.4006 

.6028 

2.4960 

.3972 

.9283  ,.9677 

10 

1.1897 

.3840 

22°  00' 

.3746 

.5736 

.4040 

.6064 

2.4751 

.3936 

.9272  .9672 

68°  00' 

1.1868 

.3869 

10 

.3773 

.5767 

.4074 

.6100 

12.4545 
12.4342 

.3900 

.9261  .9667 

50 

1.1839 

.3898 

20 

.3800 

.5798 

.4108 

.6136 

.3864 

.9250  .9661 

40 

1.1810 

.3927 

30 

.3827 

.5828 

.4142 

.6172 

2.4142 

.3828 

.9239  .9656 

30 

1.1781 

.3956 

40 

.3854 

.5859 

.4176 

.6208 

2.3945 

.3792 

.9228  .9651 

20 

1.1752 

.3985 

50 

.3881 

.5889 

.4210 

.6243 

2.3750 

.3757 

.9216  .9646 

10 

1.1723 

.4014 

23°  00' 

.,3907 

.5919 

.4245 

.6279 

2.3559 

.3721 

.9205  .9640 

67°  00' 

1.1694 

.4043 

10 

.3934 

.5948 

.4279 

.6314 

2.3369 

.3686 

.9194  .9635 

50 

1.1665 

.4072 

20 

.3961 

.5978 

.4314 

.6348 

2,3183 

.3652 

.9182  .9629 

40 

1.163(5 

.4102 

30 

.3987 

.6007 

.4348 

.6383 

2.2998 

.3617 

.9171  .9624 

30 

1.1606 

.4131 

40 

.4014 

.6036 

.4383 

.6417 

2.2817 

.3583 

.9159  .9618 

20 

1.1577 

.4160 

50 

.4041 

.6065 

.4417 

.6452 

2.2637 

.3548 

.9147  .9613 

1Q 

1.1548 

.4189 

24°  00' 

.4067 

.6093 

.4452 

.6486 

2.24(50 

.3514 

.9135  .9607 

66° 00' 

1.1519 

.4218 

10 

.4094 

.6121 

.4487 

.6520 

2.2286 

.3480 

.9124  .9602 

50 

1.1490 

.4247 

20 

.4120 

.6149 

.4522 

.6553 

2.2113 

.3447 

.9112  .9596 

40 

1.1461 

.4276 

30 

.4147 

.6177 

.4557 

.6587 

2.1943 

.3413 

.9100  .9590 

30 

1.1432 

.4305 

40 

.4173 

.6205 

.4592 

.6620 

2.1775 

.3380 

.9088  .9584 

20 

1.1403 

.4334 

50 

.4200 

.6232 

.4628 

.6654 

2.1609 

.3346 

.9075  .9579 

10 

1.1374 

.4363 

25°  00' 

.4226 

.6259 

.4663 

.6687 

2.1445 

.3313 

.9063  .9573 

65°  00' 

1.1345 

.4392 

10 

.4253 

.6286 

.4699 

.6720 

2.1283 

.32&0 

.9051  .9567 

50 

1.1316 

.4422 

20 

.4279 

.6313 

.4734 

.6752 

2.1123 

.3248 

.9038  .9561 

40 

1.1286 

.4451 

30 

.4305 

.6340 

.4770 

.6785 

2.0965 

.3215 

.9026  .9555 

30 

1.1257 

.4480 

40 

.4331 

.6366 

.4806 

.6817 

2.0809 

.3183 

.9013  .9549 

20 

1.1228 

.4509 

50 

.4358 

.6392 

.4841 

.6850 

2.0655 

.3150 

.9001  .9543 

10 

1.1199 

.4538 

26°  00' 

.4384 

.6418 

.4877 

.6882 

2.0503 

.3118 

.8988  .9537 

64°  00' 

1.1170 

.4567 

10 

.4410 

.6444 

.4913 

.6914 

2.0353 

.3086 

.8975  .9530 

50 

1.1141 

.4596 

20 

.4436 

.6470 

.4950 

.6946 

2.0204 

.3054 

.8962  .9524 

40 

1.1112 

.4625 

30 

.4462 

.6495 

.4986 

.6977 

2.0057 

.3023 

.8949  .9518 

30 

1.1083 

.4654 

40 

.4488 

.6521 

.5022 

.7009 

1.9912 

.2991 

.893(5  .9512 

20 

1.1054 

.4683 

50 

.4514 

.6546 

.5059 

.7040 

1.9768 

.2960 

.8923  .9505 

10 

1.1025 

.4712 

27°  00' 

.4540 

.6570 

.5095 

.7072 

1.9626 

.2928 

.8910  .9499 

63°  00' 

1.0996 

Value 

Logio 

Value 

LOftfl 

Value 

Loffio 

Value  '  Log10 

Degrees 

Radians 

Cosine 

Cotangent 

.  Tanoent  .     Sine 

Four  Place  Trigonometric  Functions 


115 


[Characteristi 

cs  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  value] 

Radians 

Degeees 

SlXE 

Value  Log10 

Tangent    Cotangent    Cosine 
Value  Log10  Value   Log10i  Value  Log10 

.4712 

27° 00' 

.4540  .6570 

.5095  .7072 

1.9626 

.2928 

.8910  .9499 

63°  00' 

1.0996 

.4741 

10 

.4566  .6595 

.5132  .7103 

1.94S6 

.2897 

.8897  .9492 

50 

1.0966 

.4771 

20 

.4592  .6620 

.5169  .7134 

1.9347 

.2866 

.8884  .9486 

40 

1.0937 

.4800 

30 

.4617  .6644 

.5206  .7165 

1.9210 

.2835 

.8870  .9479 

30 

1.0908 

.4829 

40 

.4643  .6668 

.5243  .7196 

1.9074 

.2804 

.8857  .9473 

20 

1.0879 

.4858 

50 

.4669  .6692 

.5280  .7226 

1.8940 

.2774 

.8843  .9466 

10 

1.0850 

.4887 

28°  00' 

.4695  .6716 

.5317  .7257 

1.8807 

.2743 

.8829  .9459 

62°  00' 

1.0821 

.4916 

10 

.4720  .6740 

.5354  .7287 

1.8676 

.2713 

.8816  .9453 

50 

1.0792 

.4945 

20 

.4746  .6763 

.5392  .7317 

1.8546 

.2683 

.8802  .9446 

40 

1.0703 

.4974 

30 

.4772  .6787 

.5430  .7348 

1.8418 

.2652 

.8788  .9439 

30 

1.0734 

,5003 

40 

.4797  .6810 

.5167  .7378 

1.8291 

.2622 

.8774  .9432 

20 

1.0705 

.5032 

50 

.4823  .6833 

.5505  .7408 

1.8165 

.2592 

.8760  .9425 

10 

1.0676 

.5061 

29°  00' 

.4848  .6856 

.5543  .7438 

1.8040 

.2562 

.8746  .9418 

61°  00' 

1.0647 

.5091 

10 

.4874  .6878 

.5581  .7467 

1.7917 

.2533 

.8732  .9411 

50 

1.0617 

.5120 

20 

.4899  .6901 

.5619  .7497 

1.7796 

.2503 

.8718  .9404 

40 

1.0688 

.5149 

30 

.4924  .6923 

.5658  .7526 

1.7675 

.2474 

.8704  .9397 

30 

1.0559 

.5178 

40 

.4950  .6946 

.5696  .7556 

1.7556 

.2444 

.8689  .9390 

20 

1.0530 

.5207 

50 

.4975  .6968 

.5735  .7585 

1.7437 

.2415 

.8675  .9383 

10 

1.0501 

.5230 

30°  00' 

.5000  .6990 

.5774  .7614 

1.7321 

.2386 

.8660  .9375 

60°  00' 

1.0472 

.5265 

10 

.5025  .7012 

.5812  .7044 

1.7205 

.2356 

.8646  .9368 

50 

1.0443 

.5294 

20 

.5050  .7033 

^5851  .7673 

1.7090 

.2327 

.8631  .9361 

40 

1.0414 

.5323 

30 

.5075  .7055 

.3890  .7701 

1.6977 

.2299 

.8616  .9353 

30 

1.0385 

.5352 

40 

.5100  .7076 

.5930  .7730 

1.6864 

.2270 

.8601  .9346 

20 

1.0356 

.5381 

50 

.5125  .7097 

.5969  .7759 

1.6753 

.2241 

.8587  .9338 

10- 

1.0327 

.5411 

31°  00' 

.5150  .7118 

.6009  .7788 

1.6643 

.2212 

.8572  .9331 

59° 00' 

1.0297 

.5440 

10 

.5175  .7139 

.6048  .7816 

1.6534 

.2184 

.8557  .9323 

50 

1.0268 

.5469 

20 

.5200  .7160 

.6088  .7845 

1.6426 

.2155 

.8542  .9315 

40 

1.0239 

.5498 

30 

.5225  .7181. 

.6128  .7873 

1.6319 

.2127 

.8526  .9308 

_30 

1.0210 

.5527 

40 

.5250  .7201 

.6168  .7902 

1.6212 

.2098 

.8511  .9300 

20 

1.0181 

.5556 

10 

.5275  .7222 

.6208  .7930 

1.6107 

.2070 

.8496  .9292 

10 

1.0152 

.5585 

32°  00' 

.5299  .7242 

.6249  .7958 

1.6003 

.2042 

.8480  .9284 

58°  00' 

1.0123 

.5(314 

10 

.5324  .7262 

.6289  .7986 

1.5900 

.2014 

.8465  .9276 

50 

1.0094 

.5643 

20 

.5318  .7282 

.6330  .8014 

1.5798 

.1986 

.8450  .9268 

40 

1.0065 

.5672 

30 

.5373  .7302 

.6371  .8042 

1.5697 

.1958 

.8434  .9260 

30 

1.0036 

.5701 

40 

.5398  .7322 

.6412  .8070 

1.5597 

.1930 

.8418  .9252 

20 

1.0007 

.5730 

50 

.5422  .7342 

.6453  .8097 

1.5497 

.1903 

.8403  .9244 

10 

.9977 

.5760 

'33°  00' 

.5446  .7361 

.6494  .8125 

1.5399 

.1875 

.8387  .9236 

57°  00' 

.9948 

.5789 

10 

.5471  .7380 

.6536  .8153 

1.5301 

.1847 

.8371  .9228 

50 

.9919 

.5818 

20 

.5495  .7400 

.6577  .8180 

1.5204 

.1820 

1^8355  .9219 
«8339  .9211 

40 

.9890 

.5847 

30 

.5519  .7419 

.6619  .8208 

1.5108 

.1792 

30 

.9861 

.5876 

40 

.5544  .7438 

.6661  .8235 

1.5013 

.1765 

.8323  .9203 

20 

.9832 

.5905 

50 

.5568  .7457 

.6703  .8263 

1.4919 

.1737 

.8307  .9194 

10 

.9803 

.5934 

34°  00' 

.5592  .7476 

.6745  .8290 

1.4826 

.1710 

.8290  .9186 

56°  00' 

.9774 

.5963 

10 

.5616  .7494 

.6787  .8317 

1.4733 

.1683 

.8274  .9177 

50 

.9745 

.5992 

20 

.5640  .7513 

.6830  .8344 

1.4641 

.1656 

.8258'  .9169 

40 

.9716 

.6021 

30 

.5664  .7531 

.6873  .8371 

1.4550 

.1629 

.8241  .9160 

30 

.9687 

.6050 

40 

.5688  .7550 

.6916  .8398 

1.4460 

.1602 

.8225  .9151 

20 

.9657 

.6080 

50 

.5712  .7568 

.6959  .8425 

1.4370 

.1575 

.8208  .9142 

10 

.9628 

.6109 

35° 00' 

.5736  .7586 

.7002  .8452 

1.4281 

.1548 

.8192  .9134 

55° 00' 

.9599 

.6138 

10 

.5760  .7604 

.7046  .8479 

1.4193 

.1521 

.8175  .9125 

50 

.9570 

.6167 

20 

.5783  .7622 

.7089  .8506 

1.4106 

.1494 

.8158  .9116 

40 

.9541 

.6196 

30 

.5807  .7640 

.7133  .8533 

1.4019 

.1467 

.8141  .9107 

30 

.9512 

.6225 

40 

.5831  .7657 

.7177  .8559 

1.3934 

.1441 

.8124  .9098 

20 

.9483 

.6254 

50 

.5854  .7675 

.7221  .8586 

1.3848 

.1414 

.8107  .9089 

10 

.9454 

.6283 

36°  00' 

.5878  .7692 

.7265  .8613 

1.3764 

.1387 

.8090  .9080 

54°  00' 

.9425 

Value  Log10 

Value  Loer10 

Value 

Log10 

Value   Log10 

Degrees 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

116  Four  Place  Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  value] 


Radians 

Degress 

Sine 

Tangent 

Cotangent 

Cosine 

Value  Log10 

Value   Log1( 

Value   Log10 

Value  Loglf 

.6283 

36°  00' 

.5878  .7602 

.7265  .8613 

1.3704  .1387 

.8090  .9080 

54°  00' 

.9425 

.6312 

10 

.5901  .7710 

.7310  .8639 

1.3080  .1301 

.8073  .9070 

50 

.9390 

.6341 

20 

.5925  .7727 

.7355  .8666 

1.3597  .1334 

,8050  .9001 

40 

.9307 

.6370 

30 

.5948  .7744 

.7400  .8692 

1.3514*  .1308 
1.34327'.1282 

[8039  .9052 

30 

.9338 

.6400 

40 

.5972  .7761 

.7445  .8718 

..8021  .9042 

20 

.9308 

.6429 

50 

.5995  .7778 

.7490  .8745 

1.3351  .1255 

.8004  .9033 

10 

.9279 

.6458 

37°  00' 

.6018  .7795 

.7536  .8771 

1.3270  .1229 

.7980  .9023 

53°  00' 

.9250 

.6487 

10 

.6041  .7811 

.7581  .8797 

1.3190  .1203 

.7909  .9014 

50 

.9221 

.6516 

20 

.6065  .7828 

.7627  .8824 

1.3111  .1170 

17951  .9004 
[7934  .8995 

40 

.9192 

.6545 

30 

.6088  .7844 

.7673  .8850 

1.3032  .1150 

30 

.9103 

.6574 

40 

.6111  .7861 

.7720  .8870 

1.2954  .1124 

.7910  .8985 

20 

.9134 

.6603 

50 

.6134  .7877 

.7766  .8902 

1.2876  .1098 

.7898  .8975 

10 

.9105 

.6632 

38°  00' 

.6157  .7893 

.7813  .8928 

1.2799  .1072 

.7880  .8905 

52°  00' 

.9070 

.6661 

10 

.6180  .7910 

.7860  .8954 

1.27231  .1046 
1:2647/ .1020 

.7802  .8955 

50 

.9047 

.6690 

20 

.6202  .7926 

,.7907  .8980 
'.7954  .9006 

.7844  .8945 

~40 

.9018 

.6720 

30 

.6225  .7941 

1.2572  .0994 

.7820  .8935 

-^30 

.8988 

.6749 
.6778 

40 
50 

.6248  .7957 
.6271  .7973 

,£002  .9032 
.8050  .9058 

1.24971  .0908 
1.24221.0942 

.7808  .8925 
.7790  .8915 

^■20 
10 

.8959 
.8930 

.6807 

39°  00' 

.6293  .7989 

.8098  .9084 

1.2349  .0910 

.7771  .8905 

51°00' 

.8901 

.6836 

10 

.6316  .8004 

.8146  .9110 

1.2270  .0890 

.7753  .8895 

50 

.8872 

.6865 

20 

.6338  .8020 

.8195  .9135 

1.2203  .0805 

.7735  .8884 

40 

.8843 

.6894 

30 

.6361  .8035 

.8243  .9161 

1.2131  .0839 

.7710  .8874 

30 

.8814 

.6923 

40 

.6383  .8050 

.8292  .9187 

1.2059  .0813 

.7098  .8804 

20 

.8785 

.6952 

50 

.6406  .8066 

.8342  .9212 

1.1988  .0788 

.7079  .8853 

10 

.8750 

.6981 

40°  00' 

.6428  .8081 

.8391  .9238 

1.1918  .0762 

.7000  .8843 

50°  00' 

.8727 

.7010 

10 

.6450  .8096 

.8441  .9264 

1.1847  .0736 

.7042  .8832 

50 

.8098 

.7039 

20 

.6472  .8111 

.8491  .9289 

1.1778  .0711 

.7023  .8821 

40 

.8008 

.7069 

30 

.6494  .8125 

.8541  .9315 

1.1708  .0685 

.7604  .8810 

30 

.8039 

.7098 

40 

.6517  .8140 

.8591  .9341 

1.1640  .0659 

.7585  .8800 

20 

.8010 

.7127 

50 

.6539  .8155 

.8642  .9366 

1.1571  .0634 

.7566  .8789 

10 

.8581 

.7156 

41°  00' 

.6561  .8169 

.8693   .9392 

1.1504  .0608 

.7547  .8778 

49°  00' 

.8552 

.7185 

10 

.6583  .8184 

.8744  .9417 

1.1436.  .0583 

.7528  .8767 

50 

.8523 

.7214 

20 

.6604  .8198 

.8796  .9443 

1.1369  .0557 

.7509  .8756 

40 

.8494 

.7243 

30 

.6626  .8213 

.8847  .9468 

1.1303  .0532 
1. 1237  .0506 

.7490  .8745 

30 

.8405 

.7272 

40 

.6648  .8227 

.8899  .9494 

.7470  .8733 

20 

.8430 

.7301 

50 

.6670  .8241 

.8952  .9519 

1.1171  .0481 

.7451  .8722 

10 

.8407 

.7330 

42°  00' 

.6691  .8255 

.9004  .9544 

1.1100  .0450 

.7431  .8711 

48°  00' 

.8378 

.7359 

10 

.6713  .8269 

.9057  .9570 

1.1041  .0430 

.7412  .8699 

50 

.8348 

.7389 

20 

.6734  .8283 

.9110  .9595 

1.0977  .0405 

.7392  .8688 

40 

.8319 

.7418 

30 

.6756  .8297 

—9163  .9021 
.9217  .9(340 

1.0913  .0379 

.7373  .8676 

,30 

.8290 

.7447 

40 

.6777  .8311 

1.0850. 0354 

.7353  .8605 

20 

.8201 

.7476 

50 

.6799  .8324 

.9271  .9071 

1.0780  .0329 

^7333  .8653 
.7314  .8641 

10 

.8232 

.7505 

43°  00' 

.6820  .8338 

.9325  .9097 

1.0724  .0303 

47° 00' 

.8203 

.7534 

10 

.6841  .8351 

.9380  .9722 

1.0001  .0278 

.7294  .8629 

50 

.8174 

.7563 

20 

.6862  .8365 

.9435  .9747 

1.0599  .0253 

.7274  .8018 

40 

.8145 

.7592 

30 

.688*^8378 

.9490  .9772 

1.0538  .0228 
1.0477  .0202 

.7254  .8000 

30 

.8110 

.7621 

40 

.6905  .8391 

.9545  .9798 

.7234  .8594 

20 

.8087 

.7650 

50 

.6926  .8405 

.9001  .9823 

1.0410  .0177 

.7214  .8582 

10 

.8058 

.7679 

44°  00' 

.6947  .8418 

.9057  .9848 

1.0355  .0152 

.7193  .8509 

46°  00' 

.8029 

.7709 

10 

.6967  .8431 

.9713  .9874 

1.0295  .0120 

.7173  .8557 

50 

.7999 

.7738 

20 

,6988  .8444 

.9770  .9899 

1.0235  .0101 

.7153  .8545 

40 

.7970 

.7767 

30 

.7009  .8457 

.9827  .9924 

1.0170  .0070 

.7133  .8532 

30 

.7941 

.7796 

40 

.7030  .8469 

.9884  .9949 

1.0117  .0051 

.7112  .8520 

20 

.7912 

.7825 

50 

.7050  .8482 

.9942  .9975 

1.0058  .0025 

.7092  .8507 

10 

.7883 

.7854 

45° 00' 

.7071  .8495 

1.0000  .0000 

1.0000  .0000 

.7071  .8495 

45°  00' 

.7854 

Value  Log10 

Value   Log10 

Value   Log10 

Value  Log10 

Degrees 

EIadians 

Cosine   { 

Cotangent 

Tangent 

Sine 

Tallies  and  Logarithms  of  Haversines 


117 


[Characteristics  of  Logarithms  omitted - 

—  determine  by  rule  from  the  value] 

0 

10' 

20' 

3C 

' 

40' 

.    5C 

' 

Value 

Log10 

Value 

Log10 

Value 

Log10 

Value 

Log10 

Value 

Log10 

Value 

Log10 

0 

.0000 

.0000  4.3254 

.0000  4.9275 

.0000  5.2796 

.0000  5.5295 

.0001  5.7233 

.0001  5.8817 

.0001  6.0156 

.0001  6.1315 

.0002 

.2338 

.0002 

.3254 

.0003 

.4081 

2 

.0003 

.4837 

.0004 

.5532 

.0004 

.6176 

.0005 

.6775 

.0005 

.7336 

.0006 

.7862 

3 

.0007 

.8358 

.0008 

.8828 

.0008 

.9273 

.0009 

.9697 

.0010 

.0101 

.0011 

.0487 

4 

.0012 

.0856 

.0013 

.1211 

.0014 

.1551 

.0015 

.1879 

.0017 

.2195 

.0018 

.2499 

5 

.0019 

.2793 

.0020 

.3078 

.0022 

.3354 

.0023 

.3621 

.0024 

.3880 

.0026 

.4132 

6 

.0027 

.4376 

.0029 

.4614 

.0031 

.4845 

.0032 

.5071 

.0034 

.5290 

.0036 

.5504 

7 

.0037 

.5713 

.0039 

.5918 

.0041 

.6117 

.0043 

.6312 

.0045 

.6503 

.0047 

.6689 

8 

.0049 

.6872 

.0051 

.7051 

.0053 

.7226 

.0055 

.7397 

.0057 

.7566 

.0059 

.7731 

9 

.0062 

.7893 

.0064 

.8052 

.0066 

.8208 

.0069 

.8361 

.0071 

.8512 

.0073 

.8660 

10 

.0076 

.8806 

.0079 

.8949 

.0081 

.9090 

.0084 

.9229 

.0086 

.9365 

.0089 

.9499 

11 

.0092 

.9631 

.0095 

.9762 

.0097 

.9890 

.0100 

.0016 

.0103 

.0141 

.0106 

.0264 

12 

.0109 

.0385 

.0112 

.0504 

.0115 

.0622 

.0119 

.0738 

.0122 

.0853 

.0125 

.0966 

13 

.0128 

.1077 

.0131 

.1187 

.0135 

.1296 

.0138 

.1404 

.0142 

.1510 

.0145 

.1614 

14 

.0149 

.1718 

.0152 

.1820 

.0156 

.1921 

.0159 

.2021 

.0163 

.2120 

.0167 

.2218 

15 

.0170 

.2314 

.0174 

.2409 

.0178 

.2504 

.0182 

.2597 

.0186 

.2689 

.0190 

.2781 

16 

.0194 

.2871 

.0198 

.2961 

.0202 

.3049 

.0206 

.3137 

.0210 

.3223 

.0214 

.3309 

17 

.0218 

.3394 

.0223 

.3478 

.0227 

.3561 

.0231 

.3644 

.0236 

.3726 

.0240 

.3806 

18 

.0245 

.3887 

.0249 

.3966 

.0254 

.4045 

.0258 

.4123 

.0263 

.4200 

.0268 

.4276 

19 

.0272 

.4352 

.0277 

.4427 

.0282 

.4502 

.0287 

.4576 

.0292 

.4649 

.0297 

.4721 

20 

.0302 

.4793 

.0307 

.4865 

.0312 

.4936 

.0317 

.5006 

.0322 

.5075 

.0327 

.5144 

21 

.0332 

.5213 

.0337 

.5281 

.0343 

.5348 

.0348 

.5415 

.0353 

.5481 

.0359 

.5547 

22 

.0364 

.5612 

.0370 

.5677 

.0375 

.5741 

.0381 

.5805 

.0386 

.5868 

.0392 

.5931 

23 

.0397 

.5993 

.0403 

.6055 

.0409 

.6116 

.0415 

.6177 

.0421 

.6238 

.0426 

.6298 

24 

.0432 

.6357 

.0438 

.6417 

.0444 

.6476 

.0450 

.6534 

.0456 

.6592 

.0462 

.6650 

25 

.0468 

.6707 

.0475 

.6764 

.0481 

.6820 

.0487 

.6876 

.0493 

.6932 

.0500 

.6987 

26 

.0506 

.7042 

.0512 

.7096 

.0519 

.7151 

.0525 

.7204 

.0532 

.7258 

.0538 

.7311 

27 

.0545 

.7364 

.0552 

.7416 

.0558 

.7468 

.0565 

.7520 

.0572 

.7572 

.0578 

.7623 

28 

.0585 

.7673 

.0592 

.7724 

.0599 

.7774 

.0606 

.7824 

.0613 

.7874 

.0620 

.7923 

29 

.0627 

.7972 

.0634 

.8020 

.0641 

.8069 

.0648 

.8117 

.0655 

.8165 

.0663 

.8213 

30 

.0670 

.8260 

.0677 

.8307 

.0684 

.8354 

.0692 

.8400 

.0699 

.8446 

.0707 

.8492 

31 

.0714 

.8538 

.0722 

.8583 

.0729 

.8629 

.0737 

.8673 

.0744 

.8718 

.0752 

.8763 

32 

.0760 

.8807 

.0767 

.8851 

.0775 

.8894 

.0783 

.8938 

.0791 

.8981 

.0799 

.9024 

33 

.0807 

.9067 

.0815 

.9109 

.0823 

.9152 

.0831 

.9194 

.0839 

.9236 

.0847 

.9277 

34 

.0855 

.9319 

.0863 

.9360 

.0871 

.9401 

.0879 

.9442 

.0888 

.9482 

.0896 

.9523 

35 

.0904 

.9563 

.0913 

.9603 

.0921 

.9643 

.0929 

.9682 

.0938 

.9722 

.0946 

.9761 

36 

.0955 

.9800 

.0963 

.9838 

.0972 

.9877 

.0981 

.9915 

.0989 

.9954 

.0998 

.9992 

37 

.1007 

.0030 

.1016 

.0067 

.1024 

.0105 

.1033 

.0142 

.1042 

.0179 

.1051 

.0216 

38 

.1060 

.0253 

.1069 

.0289 

.1078 

.0326 

.1087 

.0362 

.1096 

.0398 

.1105 

.0434 

39 

.1114 

.0470 

.1123 

.0505 

.1133 

.0541 

.1142 

.0576 

.1151 

.0611 

.1160 

.0646 

40 

.1170 

.0681 

.1179 

.0716 

.1189 

.0750 

.1198 

.0784 

.1207 

.0817 

.1217 

.0853 

41 

.1226 

.0887 

.1236 

.0920 

.1246 

.0954 

.1255 

.0987 

.1265 

.1021 

.1275 

.1054 

42 

.1284 

.1087 

.1294 

.1119 

.1304 

.1152 

.1314 

.1185 

.1323w 

.1217 

.1333 

.1249 

43 

.1343 

.1282 

.1353 

.1314 

.1363 

.1345 

.1373 

.1377 

.1383'. 1409 

.1393 

.1440 

44 

.1403 

.1472 

.1413 

.1503 

.1424 

.1534 

.1434 

.1565 

.1444 

.1596 

.1454 

.1626 

45 

.1464 

.1657 

.1475 

.1687 

.1485 

.1718 

.1495 

.1748 

.1506 

.1778 

.1516 

.1808 

46 

.1527 

.1838 

.1538 

.1867 

.1548 

.1897 

.1558 

.1926 

.1569 

.1956 

.1579 

.1985 

47 

.1590 

.2014 

.1600 

.2043 

.1611 

.2072 

.1622 

.2101 

.1633 

.2129 

.1644 

.2158 

48 

.1654 

.2186 

.1665 

.2215 

.1676 

.2243 

.1687 

.2271 

.1698 

.2299 

.1709 

.2327 

49 

.1720 

.2355 

1731 

.2382 

.1742 

.2410 

.1753 

.2437 

.1764 

.2465 

.1775 

.2492 

50 

.1786 

.2519 

.1797 

.2546 

.1808 

.2573 

.1820 

.2600 

.1831 

.2627 

.1842 

.2653 

51 

.1853 

.2680 

.1865 

.2700 

.1876 

.2732 

.1887 

.2759 

.1899 

.2785 

.1910 

.2811 

52 

.1922 

.2837 

.1933 

.2863 

.1945 

.2888 

.1956 

.2914 

.1968 

.2940 

.1979 

.2965 

53 

.1991 

.2991 

.2003 

.3016 

.2014 

.3041 

.2026 

.3066 

.2038 

:3091 

.2049 

.3116 

54 

.2061 

.3141 

.2073 

.3166 

.2085 

.3190 

.2096 

.3215 

.2108 

.3239 

.2120 

.3264 

55 

.2132 

.3288 

.2144 

.3312 

.2156 

.3336 

.2168 

.3361 

.2180 

.3384 

.2192 

.3408 

56 

.2204 

.3432 

.2216 

.3456 

.2228 

.3480 

.2240 

.3503 

.2252 

.3527 

.2265 

.3550 

57 

.2277 

.3573 

.2289 

.3596 

.2301 

.3620 

.2314 

.3643 

.2326 

.3666 

.2338 

.3689 

58 

.2350 

.3711 

.2363 

.3734 

.2375 

.3757 

.2388 

.3779 

.2400 

.3802 

.2412 

.3824 

59 

.2425 

.3847  |  .2437 

.3869 

.2450 

.3891 

.2462 

.3913 

.2475 

.3935 

.2487 

.3957 

118  Values  and  Logarithms  of  Haversines 

[Characteristics  of  Logarithms  omitted  —  determine  by  rule  from  the  value] 


60 
61 

62 
63 
64 

65 
66 
67 
68 

69 

70 
71 
72 
78 
74 

75 
76 

77 
78 
79 

80 
81 
82 
83 
84 

85 
86 
87 
88 
89 

90 
91 
92 
93 
94 

95 
96 
97 
98 
99 

100 
101 
102 
103 
104 

105 
106 
107 
108 
109 

110 
111 
112 
113 
114 

115 
116 
117 

118 
119 


0' 
Value      Log10 


.2500 
.2576 
.2653 
.2730 

.2808 

.2887 
.2966 
.3046 
.3127 
.3208 

.3290 
.3372 
.3455 
.3538 
.3622 

.3706 
.3790 
.3875 
.3960 
.4046 

.4132 

.4218 
.4304 
.4391 
.4477 

.4564 
.4651 
.4738 
.4826 
.4913 

.5000 
.5087 
.5174 
.5262 
.5349 

.5436 
.5523 
.5609 
.5696 

.5782 

.5868 
.5954 
.6040 
.6125 
.6210 

.6294 
.6378 
.6462 
.6545 
.6628 

.6710 
.6792 
.6873 
.6954 
.7034 

.7113 
.7192 
.7270 
.7347 
.7424 


.3979 
.4109 
.4237 
.4362 
.4484 

.4604 
.4722 
.4838 
.4951 
.5063 

.5172 
.5279 
.5384 
.5488 
.5589 

.5689 
.5787 
.5883 
.5977 
.6070 

.6161 
.6251 
.6339 
.6425 
.6510 

.6594 
.6676 
.6756 
.6835 
.6913 

.6990 
.7065 
.7139 
.7211 

.7283 

.7353 
.7421 

.7489 
.7556 
.7621 

.7685 
.7748 
.7810 
.7871 
.7931 

.7989 
.8047 
.8104 
.8159 
.8214 

.8267 
.8320 
.8371 

.8422 
.8472 

.8521 

.8568 
.8615 
.8661 
.8706 


10' 
Value  Log10 


.2513 
.2589 
.2665 
.2743 
.2821 

.2900 
.2980 
.3060 
.3140 
.3222 

.3304 
.3386 
.3469 
.3552 
.3636 

.3720 
.3805 
.3889 
.3975 
.4060 

.4146 
.4232 
.4319 
.4405 
.4492 

.4579 


.4753 
.4840 
.4937 

.5015 
.5102 
.5189 
.5276 
.5363 

.5450 
.5537 
.5624 
.5710 
.5797 

.5883 
.5968 
.6054 
.6139 
.6224 

.6308 
.6392 
.6476 
.6559 
.6642 

.6724 
.6805 

.6887 
.6967 
.7047 

.7126 
.7205 
.7283 
.7360 
.7437 


.4001 
.4131 
.4258 
.4382 
.4504 

.4624 
.4742 

.4857 
.4970 
.5081 

.5190 
.5297 
.5402 
.5505 
.5606 

.5705 
.5803 
.5899 
.5993 
.6085 

.6176 
.6266 
.6353 
.6440 
.6524 

.6607 
.6689 
.6770 
.6848 
.6926 

.7002 
.7077 
.7151 
.7223 
.7294 

.7364 
.7433 
.7500 
.7567 
.7632 

.7696 
.7759 

.7820 
.7881 
.7940 

.7999 
.8056 
.8113 
.8168 
.8223 

.8276 
.8329 
.8380 
.8430 
.8480 

.8529 

.8576 
.8623 
.8669 
.8714 


20' 
Value  Logt 


30' 
Value  Log10 


.2525  .4023 
.2601  .4152 
.2678  .4279 
.2756  .4403 
.2834  .4524 

.2913  .4644 
.2993  .4761 
.3073  .4876 
.3154  .4989 
.3235  .5099 

.3317  .5208 

.3400  .5314 

.3483  .5419 

.3566  .5522 

.3650  .5623 

.3734  .5722 

.3819  .5819 

.3904  .5915 

.3989  .6009 

.4075  .6101 

.4160  .6191 
.4247  .6280 
.4333  .6368 
.4420  .6454 
.4506  .6538 

.4593  .6621 
.4680  .6703 
.4767  .6783 
.4855  .6862 
.4942  .6939 

.5029  .7015 
.5116  .7090 
.5204  .7163 
.5291  .7235 
.5378  .7306 


.5465 
.5552 
.5638 
.5725 
.5811 

.5897 
.5983 
.6068 
.6153 
.6238 

.6322 
.6406 
.6490 
.6573 
.6655 

.6737 
.6819 
.6900 
,6980 
.7060 

.7139 

.7218 
.7296 
.7373 
.7449 


.7376 
.7444 
.7511 

.7577 
.7642 

.7706 
.7769 
.7830 
.7891 
.7950 

.8009 


.8122 
.8177 
.8232 

.8285 
.8337 
.8388 
.8439 

.8488 

.8537 

.8584 
.8631 
.8676 
.8721 


.2538 
.2614 
.2691 
.2769 

.2847 

.2927 
.3006 
.3087 
.3167 
.3249 

.3331 
.3413 
.3496 
.3580 
.3664 

.3748 
.3833 
.3918 
.4003 
.4089 

.4175 
.4261 
.4347 
.4434 
.4521 

.4608 
.4695 
.4782 
.4869 
.4956 

.5044 
.5131 
.5218 
.5305 
.5392 

.5479 
.5566 
.5653 
.5739 
.5825 

.5911 
.5997 
.6082 
.6167 
.6252 

.6336 
.6420 
.6504 
.6587 
.6669 

.6751 

.6833 
.6913 
.6994 
.7073 

.7153 
.7231 
.7309 
.7386 
.7462 


.4045 
.4173 
.4300 
.4423 
.4545 

.4664 
.4780 
.4895 
.5007 
.5117 

.5226 
.5332 
.5436 
.5539 
.5639 

.5738 
.5835 
.5930 
.6024 
.6116 

.6206 
.6295 
.6382 
.6468 
.6552 

.6635 
.6716 
.6796 
.6875 
.6952 

.7027 
.7102 
.7175 
.7247 
.7318 

.7387 
.7455 
.7523 
.7588 
.7653 

.7717 

.7779 
.7841 
.7901 
.7960 

.8018 
.8075 
.8131 

.8187 
.8241 

.8294 
.8346 
.8397 

.8447 
.8496 

.8545 
.8592 
.8638 
.8684 
.8729 


40' 
Value      Log10 


.2551 
.2627 
.2704 

.2782 
.2861 

.2940 
.3020 
.3100 
.3181 
.3263 

.3345 
.3427 
.3510 
.3594 
.3678 

.3762 
.3847 
.3932 
.4017 
.4103 

.4189 
.4275 
.4362 
.4448 
.4535 

.4622 
.4709 
.4796 
.4884 
.4971 

.5058 
.5145 
.5233 
.5320 
.5407 

.5494 
.5580 
.5667 
.5753 
.5840 

.5925 
.6011 
.6096 
.6181 
.6266 

.6350 
.6434 
.6517 
.6600 

.6683 

.6765 
.6846 
.6927 
.7007 

.7087 

.7166 
.7244 
.7322 
.7399 
.7475 


.4066 
.4195 
.4320 
.4444 
.4565 

.4683 
.4799 
.4914 
.5026 
.5136 

.524*4 
.5349 
.5454 
.5556 
.5656 

.5754 
.5851 
.5946 
.6039 
.6131 

.6221 
.6310 
.6397 
.6482 
.6566 

.6649 
.6730 
.6809 
.6887 
.6964 

.7040 
.7114 
.7187 
.7259 
.7329 

.7399 

.7467 
.7534 
.7599 
.7664 

.7727 
.7790 
.7851 
.7911 
.7970 

.8028 
.8085 
.8141 
.8196 
.8250 

.8302 
.8354 
.8405 
.8455 
.8504 

.8553 
.8600 
.8646 
.8691 
.8736 


50' 
Value      Log10 


.2563 
.2640 
.2717 
.2795 

.2874 

.2953 
.3033 
.3113 
.3195 
.3276 

.3358 
.3441 
.3524 
.3608 
.3692 

.3776 
.3861 
.3946 
.4032 
.4117 

.4203 
.4290 
.4376 
.4463 
.4550 

.4637 
.4724 
.4811 
.4898 
.4985 

.5073 
.5160 
.5247 
.5334 
.5421 

.5508 
.5595 
.5682 
.5768 
.5854 

.5940 
.6025 
.6111 
.6195 
.6280 

.6364 
.6448 
.6531 
.6614 
.6696 

.6778 
.6860 
.6940 
.7020 
.7100 

.7179 

.7257 
.7335 
.7411 

.7487 


.4088 
.4216 
.4341 
.4464 
.4584 

.4703 
.4819 
.4932 
.5044 
.5154 

.5261 
.5367 
.5471 
.5572 
.5672 

.5771 

.5867 
.5962 
.6055 
.6146 

.6236 
.6324 
.6411 
.6496 
.6580 

.6662 
.6743 
.6822 
.6900 
.6977 

.7052 
.7126 
.7199 
.7271 
.7341 

.7410 

.7478 
.7545 
.7610 
.7674 

.7738 
.7800 
.7861 
.7921 
.7980 

.8037 
.8094 
.8150 
.8205 

.8258 

.8311 
.8363 
.8414 
.8464 
.8513 
.8561 
.8608 
.8654 
.8699 
.8743 


Values  and  Logarithms  of  Haversines 

[Characteristics  of  Logarithms  omitted  —  determine  by  rule  from  the  value] 


119 


. 

C 

10' 

20' 

30' 

40' 

5)' 

Value 

Logw 

Value 

Logi0 

Value 

Log10 

Value 

Log10 

Value 

Log10 

Value 

Log10 

120 

.7500 

.8751 

.7513 

.8758 

.7525 

.8765 

.7538 

.8772 

.7550 

.8780 

.7563 

.8787 

121 

.7575 

.8794 

.7588 

.8801 

.7600 

.8808 

.7612 

.8815 

.7625 

.8822 

.7637 

.8829 

122 

.7650 

.8836 

.7662 

.8843 

.7674 

.8850 

.7686 

.8857 

.7699 

.8864 

.7711 

.8871 

123 

.7723 

.8878 

.7735 

.8885 

.7748 

.8892 

.7760 

.8898 

.7772 

.8905 

.7784 

.8912 

124 

.7796 

.8919 

.7808 

.8925 

.7820 

.8932 

.7832 

.8939 

.7844 

.8945 

.7856 

.8952 

125 

.7868 

.8959 

.7880 

.8965 

.7892 

.8972 

.7904 

.8978 

.7915 

.8985 

.7927 

.8991 

126 

.7939 

.8998 

.7951 

.9004 

.7962 

.9010 

.7974 

.9017 

.7986 

.9023 

.7997 

.9030 

127 

.8009 

.9036 

.8021 

.9042 

.8032 

.9048 

.8044 

.9055 

.8055 

.9061 

.8067 

.9067 

128 

.8078 

.9073 

.8090 

.9079 

.8101 

.9085 

.8113 

.9092 

.8124 

.9098 

.8135 

.9104 

129 

.8147 

.9110 

.8158 

.9116 

.8169 

.9122 

.8180 

.9128 

.8192 

.9134 

.8203 

.9140 

130 

.8214 

.9146 

.8225 

.9151 

.8236 

.9157 

.8247 

.9163 

.8258 

.9169 

.8269 

.9175 

131 

.8280 

.9180 

.8291 

.9186 

.8302 

.9192 

.8313 

.9198 

.8324 

.9203 

.8335 

.9209 

132 

.8346 

.9215 

.8356 

.9220 

.8367 

.9226 

.8378 

.9231 

.8389 

.9237 

.8399 

.9242 

133 

.8410 

.9248 

.8421 

.9253 

.8431 

.9259 

.8442 

.9264 

.8452 

.9270 

.8463 

.9275 

134 

.8473 

.9281 

.8484 

.9286 

.8494 

.9291 

.8501 

.9297 

.8515 

.9302 

.8525 

.93p7 

135 

.8536 

.9312 

.8546 

.9318 

.8556 

.9323 

.8566 

.9328 

.8576 

.9333 

.8587 

.9338 

136 

.8597 

.9343 

.8607 

.9348 

.8617 

.9353 

.8627 

.9359 

.8637 

.9364 

.8647 

.9369 

137 

.8657 

.9374 

.8667 

.9379 

.8677 

.9383 

.8686 

.9388 

.8696 

.9393 

.8706 

.9398 

138 

.8716 

.9403 

.8725 

.9408 

.8735 

.9413 

.8745 

.9417 

.8754 

.9422 

.8764 

.9427 

139 

.8774 

.9432 

.8783 

.9436 

.8793 

.9441 

.8802 

.9446 

.8811 

.9450 

.8821 

.9455 

140 

.8830 

.9460 

.8840 

.9464 

.8849 

.9469 

.8858 

.9473 

.8867 

.9478 

.8877 

.9482 

141 

.8886 

.9487 

.8895 

.9491 

.8904 

.9496 

.8913 

.9500 

.8922 

.9505 

.8931 

.9509 

142 

.8940 

.9513 

.8949 

.9518 

.8958 

.9522 

.8967 

.9526 

.8976 

.9531 

.8984 

.9535 

143 

.8993 

.9539 

.9002 

.9543 

.9011 

.9548 

.9019 

.9552 

.9028 

.9556 

.9037 

.9560 

144 

.9045 

.9564 

.9054 

.9568 

.9062 

.9572 

.9071 

.9576 

.9079 

.9580 

.9087 

.9584 

145 

.9096 

.9588 

.9104 

.9592 

.9112 

.9596 

.9121 

.9600 

.9129 

.9604 

.9137 

.9608 

146 

.9145 

.9612 

.9153 

.9616 

.9161 

.9620 

.9169 

.9623 

.9177 

.9627 

.9185 

.9631 

147 

.9193 

.9635 

.9201 

.9638 

.9209 

.9642 

.9217 

.9646 

.9225 

.9650 

.9233 

.9653 

148 

.9240 

.9657 

.9248 

.9660 

.9256 

.9664 

.9263 

.9668 

.9271 

.9671 

.9278 

.9675 

149 

.9286 

.9678 

.9293 

.9682 

.9301 

.9685 

.9308 

.9689 

.9316 

.9692 

.9323 

.9695 

150 

.9330 

.9699 

.9337 

.9702 

.9345 

.9706 

.9352 

.9709 

.9359 

.9712 

.9366 

.9716 

151 

.9373 

.9719 

.9380 

.9722 

.9387 

.9725 

.9394 

.9729 

.9401 

.9732 

.9408 

.9735 

152 

.9415 

.9738 

.9422 

.9741 

.9428 

.9744 

.9435 

.9747 

.9442 

.9751 

.9448 

.9754 

153 

.9455 

.9757 

.9462 

.9760 

.9468 

.9763 

.9475 

.9766 

.9481 

.9769 

.9488 

.9772 

154 

.9494 

.9774 

.9500 

.9777 

.9507 

.9780 

.9513 

.9783 

.9519 

.9786 

.9525 

.9789 

155 

.9532 

.9792 

.9538 

.9794 

.9544 

.9797 

.9550 

.9800 

.9556 

.9803 

.9562 

.9805 

156 

.9568 

.9808 

.9574 

.9811 

.9579 

.9813 

.9585 

.9816 

.9591 

.9819 

.9597 

.9821 

157 

.9603 

.9824 

.9608 

.9826 

.9614 

.9829 

.9619 

.9831 

.9625 

.9834 

.9630 

.9836 

158 

.9636 

.9839 

.9641 

.9841 

.9647 

.9844 

.9652 

.9846 

.9657 

.9849 

.9663 

.9851 

159 

.9668 

.9853 

.9673 

.9856 

.9678 

.9858 

.9683 

.9860 

.9688 

.9863 

.9693 

.9865 

160 

.9698 

.9867 

.9703 

.9869 

.9708 

.9871 

.9713 

.9874 

.9718 

.9876 

.9723 

.9878 

161 

.9728 

.9880 

.9732 

.9882 

.9737 

.9884 

.9742 

.9886 

.9746 

.9888 

.9751 

.9890 

162 

.9755 

.9892 

.9760 

.9894 

.9764 

.9896 

.9769 

.9898 

.9773 

.9900 

.9777 

.9902 

163 

.9782 

.9904 

.9786 

.9906 

.9790 

.9908 

.9794 

.9910 

.9798 

.9911 

.9802 

.9913 

164 

.9806 

.9915 

.9810 

.9917 

.9814 

.9919 

.9818 

.9920 

.9822 

.9922 

.9826 

.9923 

165 

.9830 

.9925 

.9833 

.9927 

.9837 

.9929 

.9841 

.9930 

.9844 

.9932 

.9848 

.9933 

166 

.9851 

.9935 

.9855 

.9937 

.9858 

.9938 

.9862 

.9940 

.9865 

.9941 

.9869 

.9943 

167 

.9872 

.9944 

.9875 

.9945 

.9878 

.9947 

.9881 

.9948 

.9885 

.9950 

.9888 

.9951 

168 

.9891 

.9952 

.9894 

.9954 

.9897 

.9955 

.9900 

.9956 

.9903 

.9957 

.9905 

.9959 

169 

.9908 

.9960 

.9911 

.9961 

.9914 

.9962 

.9916 

.9963 

.9919 

.9965 

.9921 

.9966 

170 

.9924 

.9967 

.9927 

.9968 

.9929 

.9969 

.9931 

.9970 

.9934 

.9971 

.9936 

.9972 

171 

.9938 

.9973 

.9941 

.9974 

.9943 

.9975 

.9945 

.9976 

.9947 

.9977 

.9949 

9978 

172 

.9951 

.9979 

.9953 

.9980 

.9955 

.9981 

.9957 

.9981 

.9959 

.9982 

.9961 

.9983 

173 

.9963 

.9984 

.9964 

.9984 

.9966 

.9985 

.9968 

.9986 

.9969 

.9987 

.9971 

.9987 

174 

.9973 

.9988 

.9974 

.9988 

.9976 

.9989 

.9977 

.9990 

.9978 

.9991 

.9980 

.9991 

175 

.9981 

.9992 

.9982 

.9992 

.9983 

.9993 

.9985 

.9993 

.9986 

.9994 

.9987 

.9994 

176 

.9988 

.9995 

.9989 

.9995 

.9990 

.9996 

.9991 

.9996 

.9992 

.9996 

.9992 

.9997 

177 

.9993 

.9997 

.9994 

.9997 

.9995 

.9998 

.9995 

.9998 

.9996 

.9998 

.9996 

.9998 

178 

.9997 

.9999 

.9997 

.9999 

.9998 

.9999 

.9998 

.9999 

.9999 

.9999 

.9999 

.9999 

179 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999 

.9999  0.0000 

1.0000 

.0000 

INDEX 


Abscissa,  6. 

Absolute  value,  of  a  directed  quan- 
tity, 7. 

Addition,  of  angles,  9;  formulas  in 
trigonometry,  95. 

Angle,  definition  of,  7;  directed,  7; 
measurement  of,  8 ;  addition  and 
subtraction  of,  9  ;  functions  of,  2  ; 
of  elevation  and  depression,  16; 
of  triangle,  48 ;  in  artillery  service, 
76. 

Annuities,  70. 

Arc  of  a  circle,  76. 

Artillery  service,  use  of  angles  in,  76. 

Axes,  of  coordinates,  5. 

Briggian  logarithms,  54. 

Characteristic  of  a  logarithm,  54. 
Cologarithms,  59. 
Common  logarithms,  54. 
Compass,  Mariner's,  29. 
Computation,     numerical, 

logarithmic,  61  ff. 
Coordinates  in  a  plane,  5. 
Cosecant,  32. 
Cosine,  definition  of,   12 : 

of,  81 ;    graph  of,  82  ; 

40. 
Cotangent,  definition  of,  32 
Course,  29. 
Coversed  sine,  32. 


18,     24 


;     variation 
law  of  — s, 


Dead  reckoning,  30. 

Departure,  29. 

Difference  in  latitude,  29 ;  in  longi- 
tude, 30. 

Directed,  angles,  7  ;  quantities,  6 
segments,  7. 

Distance,  29. 

Elements  of  a  triangle,  1. 


Function,  definition  of,  3 ;  representa- 
tion of,  32  ;     trigonometric,    12  ff ., 

58. 

Graph  of  trigonometric  functions, 
80,  82,  83. 

Haversine,  definition  of,  32;  solu- 
tion of  triangles  by,  48 ;  tables  of, 
117-9. 

Identities,  trigonometric,  35. 
Initial  position,  7. 
Interest,  70. 
Interpolation,  22. 


Knot,  29. 

Latitude,  difference  in,  29 ;  middle, 
30. 

Law,  of  sines,  40 ;  cosines,  40 ;  of 
tangents,  47. 

Logarithm,  definition  of,  52 ;  inven- 
tion of,  50 ;  laws  of,  53  ;  systems 
of,  54 ;  characteristic  and  man- 
tissa of,  54 ;  use  of  tables  of,  56 ; 
tables  of,  110-16. 

Logarithmic  scale,  73. 

Magnitude,  6. 
Mantissa,  54. 
Mariner's  compass,  29. 
Middle  latitude,  30. 
Mil,  76. 

Napier,  J.,  50. 
Nautical  mile,  29. 
Navigation,  28  ff . 

Negative  angle,  definition  of,  7; 
functions  of,  85. 

Ordinate,  6. 

121 


122 


INDEX 


Parts  of  a  triangle,  1. 

Period    of    trigonometric    functions, 

80,  82,  84. 
Plane  sailing,  28. 
Plane  trigonometry,  1. 
Product  formulas,  101. 
Projectile,  72. 
Projection,  92. 

Quadrant,  6. 

Radian,  75. 

Radius  of  inscribed  circle,  46. 
Rotation,  angles  of,  8. 
Rounded  numbers,  25. 

Scale,  logarithmic,  73. 

Secant,  definition  of,  32. 

Significant  figures,  25. 

Sine,  definition  of,  12 ;    variation  of, 

79  ;  graph  of,  80 ;  law  of s,  40. 

Slide  rule,  74. 


Solution  of  triangles,   1,   16  ff.,  41  ft*., 

48,  62  ff. 
Spherical  trigonometry,  1. 

Tables,  of  squares,  27,  106-7;  of 
haversines,  117-9;  of  logarithms, 
110-11 ;  of  trigonometric  func- 
tions, 112-19. 

Tangent,  definition  of,  3, 12  ;  variation 
of,  82 ;  graph  of,  83 ;  line  repre- 
sentation of,  83 ;  law  of s,  47. 

Triangle,  area  of,  45  ;  angles  of,  48 ; 
solution  of,  1,  16  ff.,  41  ff.,  48,  62. 

Trigonometric  equations,  88. 

Trigonometric  functions,  definitions 
of,  3,  12,  15,  32 ;  graphs  of,  80,  82, 
83 ;  computation  of,  18  ff . ;  periods 
of,  80,  82,  84;  inverse,  87;  formulas, 
15,  32,  34,  96  ff. ;  logarithms  of, 
61 ;  tables  of,  21,  112-19. 

Versed  sine,  defined,  32. 


Printed  in  the  United  States  of  America. 


\})     ^    -.    At-  V 

(.^       -]   ^    <Un>  3-f-  -    Aw 


h   i~    \ 


V 


w*. 


.  ^  a-*,  ^  £  6   f^U^-W^J 


T^HE  following  pages  contain  advertisements  of  a 
few  of  the  Macmillan  books  on  kindred  subjects. 


ELEMENTARY  MATHEMATICAL 
ANALYSIS 

BY 

JOHN  WESLEY  YOUNG 

Professor  of  Mathematics  in  Dartmouth  College 

And  FRANK  MILLET  MORGAN 

Assistant  Professor  of  Mathematics  in  Dartmouth  College 


Edited   by   Earle    Raymond    Hedrick,    Professor  of  Mathematics 
in   the   University   of  Missouri 

77/.,  Cloth,  i2tno,  $2.60 

1  .  A  textbook  for  the  freshman  year  in  colleges,  universities,  and 
technical  schools,  giving  a  unified  treatment  of  the  essentials  of 
trigonometry,  college  algebra,  and  analytic  geometry,  and  intro- 
ducing the  student  to  the  fundamental  conceptions  of  calculus. 

The  various  subjects  are  unified  by  the  great  centralizing 
theme  of  functionality  so  that  each  subject,  without  losing  its 
fundamental  character,  is  shown  clearly  in  its  relationship  to  the 
others,  and  to  mathematics  as  a  whole. 

More  emphasis  is  placed  on  insight  and  understanding  of 
fundamental  conceptions  and  modes  of  thought ;  less  emphasis 
on  algebraic  technique  and  facility  of  manipulation.  Due  recog- 
nition is  given  to  the  cultural  motive  for  the  study  of  mathe- 
matics and  to  the  disciplinary  value. 

The  text  presupposes  only  the  usual  entrance  requirements  in 
elementary  algebra  and  plane  geometry. 


THE   MACMILLAN   COMPANY 

Publishers  64-66  Fifth  Avenue  New  York 


Trigonometry 


By  ALFRED   MONROE   KENYON 

Professor  of  Mathematics,  Purdue  University 

and 

LOUIS   INGOLD 

Assistant  Professor  of  Mathematics,  the  University  of  Missouri 

Edited  by  Earle  Raymond  Hedrick 

With  Brief  Tables,  8vo,  $1.20 
With  Complete  Tables,  8vo,  $1.50 

The  book  contains  a  minimum  of  purely  theoretical  matter.  Its  entire  organization  is 
intended  to  give  a  clear  view  of  the  meaning  and  the  immediate  usefulness  of  Trigonometry. 
The  proofs,  however,  are  in  a  form  that  will  not  require  essential  revision  in  the  courses  that 
follow.  .  .  . 

The  number  of  exercises  is  very  large,  and  the  traditional  monotony  is  broken  by  illus- 
trations from  a  variety  of  topics.  Here,  as  well  as  in  the  text,  the  attempt  is  often  made  to 
lead  the  student  to  think  for  himself  by  giving  suggestions  rather  than  completed  solutions 
or  demonstrations. 

The  text  proper  is  short;  what  is  there  gained  in  space  is  used  to  make  the  tables  very 
complete  and  usable.  Attention  is  called  particularly  to  the  complete  and  handily  arranged 
table  of  squares,  square  roots,  cubes,  etc. ;  by  its  use  the  Pythagorean  theorem  and  the  Cosine 
Law  become  practicable  for  actual  computation.  The  use  of  the  slide  rule  and  of  four-place 
tables  is  encouraged  for  problems  that  do  not  demand  extreme  accuracy. 

Analytic  Geometry  and  Principles  of  Algebra 

By  ALEXANDER  ZIWET 

Professor  of  Mathematics,  the  University  of  Michigan 

and 

LOUIS  ALLEN  HOPKINS 

Instructor  in  Mathematics,  the  University  of  Michigan 

Edited  by  Earle  Raymond  Hedrick 

Cloth,  i2tno,  $1.75 

This  work  combines  with  analytic  geometry  a  number  of  topics  traditionally  treated  in 
college  algebra  that  depend  upon  or  are  closely  associated  with  geometric  sensation.  Through 
this  combination  it  becomes  possible  to  show  the  student  more  directly  the  meaning  and  the 
usefulness  of  these  subjects. 

The  treatment  of  solid  analytic  geometry  follows  the  more  usual  lines.  But,  in  view  of  the 
application  to  mechanics,  the  idea  of  the  vector  is  given  some  prominence;  and  the  represen- 
tation of  a  function  of  two  variables  by  contour  lines  as  well  as  by  a  surface  in  space  is  ex- 
plained and  illustrated  by  practical  examples. 

The  exercises  have  been  selected  with  great  care  in  order  not  only  to  furnish  sufficient 
material  for  practice  in  algebraic  work  but  also  to  stimulate  independent  thinking  and  to 
point  out  the  applications  of  the  theory  to  concrete  problems.  The  number  of  exercises  is 
sufficient  to  allow  the  instructor  to  make  a  choice. 


THE   MACMILLAN   COMPANY 

Publishers  64-66  Fifth  Avenue  New  York 


A  Short  Course  in  Mathematics 

By  R.   E.   MORITZ 
Professor  of  Mathematics,  University  of  Washington 

Cloth,  i2tno 

A  text  containing  the  material  essential  for  a  short  course  in  Freshman  Mathematics  which 
is  complete  in  itself,  and  which  contains  no  more  material  than  the  average  Freshman  can 
assimilate.  The  book,  will  constitute  an  adequate  preparation  for  further  study,  and  will 
enable  the  student  to  take  up  the  usual  course  in  analytical  geometry  without  any  handicap. 

Among  the  subjects  treated  are :  Factoring,  Radicals,  Fractional  and  Negative  Exponents, 
Imaginary  Quantities,  Linear  and  Quadratic  Equations ;  Coordinates,  Simple  and  Straight 
Line  Graphs,  Curve  Plotting,  Maxima  and  Minima,  Areas;  The  General  Angle  and  Its 
Measures,  The  Trigonometric  or  Circular  Functions,  Functions  of  an  Acute  Angle;  Solution 
of  Right  and  Oblique  Triangles;  Exponents  and  Logarithms;  Application  of  Logarithms  to 
Numerical  Exercises,  to  Mensuration  of  Plane  Figures,  and  to  Mensuration  of  Solids;  The 
Four  Cases  of  Oblique  Triangles,  Miscellaneous  Problems  Involving  Triangles. 

Plane  and  Spherical  Trigonometry 

By  LEONARD   M.   PASSANO 

Associate  Professor  of  Mathematics  in   the  Massachusetts  Institute  of 

Technology 

Cloth,  8vo,  $1.25 

The  chief  aims  of  this  text  are  brevity,  clarity,  and  simplicity.  The  author  presents  the 
whole  field  of  Trigonometry  in  such  a  way  as  to  make  it  interesting  to  students  approaching 
some  maturity,  and  so  as  to  connect  the  subject  with  the  mathematics  the  student  has  pre- 
viously studied  and  with  that  which  may  follow. 

CONTENTS 

PLANE  TRIGONOMETRY  chapter 

6.   The  Solution  of  General  Triangles     .     . 


The   Solution   of  Trigonometric   Equa- 


CHAPTER 

1.  The  Trigonometric  Functions  of  Any  tions 

Angle  and  Identical  Relations  among 
Them 

2.  Identical  Relations  Among  the  Func-  SPHERICAL  TRIGONOMETRY 

tions  of  Related  Angles:     The  Values 

of  the  Functions  of  Certain  Angles  8.   Fundamental  Relations 

3.  The     Solution     of     Right     Triangles.  9.   The   Solution  of   Right   Spherical  Tri 

Logarithms    and     Computation     by  angles 

Means  of  Logarithms 10.   The     Solution    of    Oblique     Spherical 

4.  Fundamental  Identities       Triangles 

5.  The  Circular  or  Radian  Measure  of  an  n.  The  Earth  as  a  Sphere                    ... 

Angle.     Inverse  Trigonometric  Func-        Answers 

tions 


THE   MACMILLAN   COMPANY 

Publishers  64-66  Fifth  Avenue  New  York 


Differential  and  Integral  Calculus 

By  CLYDE  E.  LOVE,   Ph.D. 

Assistant  Professor  of  Mathematics  in  the  University  of  Michigan 

Crown  8vo,  $2.10 

Presents  a  first  course  in  the  calculus  —  substantially  as  the  author  has 
taught  it  at  the  University  of  Michigan  for  a  number  of  years.  The  follow- 
ing points  may  be  mentioned  as  more  or  less  prominent  features  of  the  book  : 

In  the  treatment  of  each  topic  the  author  has  presented  his  material  in 
such  a  way  that  he  focuses  the  student's  attention  upon  the  fundamental 
principle  involved,  insuring  his  clear  understanding  of  that,  and  preventing 
him  from  being  confused  by  the  discussion  of  a  multitude  of  details.  His 
constant  aim  has  been  to  prevent  the  work  from  degenerating  into  mere 
mechanical  routine;  thus,  wherever  possible,  except  in  the  purely  formal 
parts  of  the  course,  he  has  avoided  the  summarizing  of  the  theory  into 
rules  or  formulas  which  can  be  applied  blindly. 

The  Calculus 

By  ELLERY  WILLIAMS  DAVIS 

Professor  of  Mathematics,  the  University  of  Nebraska 

Assisted  by  William  Charles  Brenke,  Associate  Professor  of  Mathe- 
matics, the  University  of  Nebraska 

Edited  by  Earle  Raymond  Hedrick 

Cloth,  semi- flexible,  with  Tables,  i2tno,  $2.10 
Edition  De  Luxe,  flexible  leather  binding,  $2.50 

This  book  presents  as  many  and  as  varied  applications  of  the  Calculus 
as  it  is  possible  to  do  without  venturing  into  technical  fields  whose  subject 
matter  is  itself  unknown  and  incomprehensible  to  the  student,  and  without 
abandoning  an  orderly  presentation  of  fundamental  principles. 

The  same  general  tendency  has  led  to  the  treatment  of  topics  with  a  view 
toward  bringing  out  their  essential  usefulness.  Rigorous  forms  of  demon- 
stration are  not  insisted  upon,  especially  where  the  precisely  rigorous  proofs 
would  be  beyond  the  present  grasp  of  the  student.  Rather  the  stress  is  laid 
upon  the  student's  certain  comprehension  of  that  which  is  done,  and  his  con- 
viction that  the  results  obtained  are  both  reasonable  and  useful.  At  the 
same  time,  an  effort  has  been  made  to  avoid  those  grosser  errors  and  actual 
misstatements  of  fact  which  have  often  offended  the  teacher  in  texts  other- 
wise attractive  and  teachable. 

THE   MACMILLAN   COMPANY 

Publishers  64-66  Fifth  Avenue  New  Tork 


S~-    f  (   l+L-)^ 


^-Cv, 


'• 


""""■.SSSHSy-."™ 


10w-7. 


Iir,i 


YB 


Y/.o9(.1 


889757(^^533 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


(,.o  1  rc 


